Wikiversity:Colloquium

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== Requested update to [[Wikiversity:Interface administrators]] ==

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

:And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC)
::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:46, 20 August 2025 (UTC)
:I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:57, 2 December 2025 (UTC)
::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC)
:@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC)
::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 25 January 2026 (UTC)
:::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC)
:I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC)
:: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC)
:I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:47, 13 April 2026 (UTC)
::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC)

== Curators and curators policy ==

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

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

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

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

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

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

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

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

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

:There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC)
:It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC)

== Inactivity policy for Curators ==

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

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

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

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

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

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

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

:And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them &quot;breaks out of the chain&quot;, the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC)
::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC)
: @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC)

== Technical Request:  Courtesy link.. ==

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

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

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

== WikiEducator has closed ==

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

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

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

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

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

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

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

== Wikinews is ending ==

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

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

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

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

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

Hello everyone,

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

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

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

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

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

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

== Enable the abuse filter block action? ==

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Wikiversity:Curators to become a policy ==

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

:*{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)</text>
      <sha1>fksfghbdiogdvkatopyai95f4lgd4u1</sha1>
    </revision>
  </page>
  <page>
    <title>Hello, world!</title>
    <ns>0</ns>
    <id>2103</id>
    <revision>
      <id>2805792</id>
      <parentid>2762168</parentid>
      <timestamp>2026-04-21T15:58:04Z</timestamp>
      <contributor>
        <username>~2026-24486-77</username>
        <id>3067774</id>
      </contributor>
      <comment>/* JavaScript (aka JScript, ECMAScript, LiveScript) */</comment>
      <origin>2805792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13732" sha1="e2efveprauzn7dh07skm17qp8bbgomn" xml:space="preserve">[[File:Hello World Brian Kernighan 1978.jpg|thumb|right|Hello World! by Brian Kernighan. Based on a 1978 Bell Laboratories internal memorandum by Brian Kernighan, Programming in C: A Tutorial, which contains the first known version.]]

As described in more detail in [[w:&quot;Hello,_World!&quot;_program|the related Wikipedia article]], '''Hello, world!''' is a classic &quot;first program&quot; one creates when learning a new programming language. The objective of the application is the same: to print the text &quot;Hello, world!&quot; to the screen in some form, be it console output or a dialog.

In many cases, the statement required to do this is just a single line.

It seems appropriate that our introduction to Computer Science occupied this title. As a student, the first choice to make is to decide ''what kind of knowledge you are looking for''. Of course, this depends upon your needs. You might be:

* A learned computer scientist or professional eager to contribute research and course material
* Computer professional seeking an alternative to expensive commercial certification
* Adult non-computer professional or entrepreneur who could benefit from academic/practical knowledge of computing
* College-eligible (or not) student considering a degree
* Casual user trying to to catch/spread the next [[w:Computer virus|virus]]
* Hobbyist or computer gamer looking to get the most out of your computing experience
* Complete newbie looking for a place to start

This is an exciting time for education, and for those of us wishing to collaborate and share knowledge, skills and experience. At present, we are only limited by the sky, and some very large hard drives in a server farm somewhere.

{{TOC right}}

== Examples of ''Hello, world!'' ==

=== [[Hello World/Ada|Ada]] ===
&lt;syntaxhighlight lang=&quot;ada&quot;&gt;

 
 procedure Hello is
 begin
    Ada.Text_IO.Put_Line (&quot;Hello, world!&quot;);

 with Ada.Text_IO;
&lt;/syntaxhighlight&gt;
For an explanation see [[wikibooks:Ada Programming:Basic|b:Ada Programming:Basic]].

=== ASP ===
&lt;syntaxhighlight lang=&quot;asp&quot;&gt;
&lt;%
    Response.Write &quot;Hello, world!&quot;
%&gt;
&lt;/syntaxhighlight&gt;

or

&lt;syntaxhighlight lang=&quot;asp&quot; line=&quot;1&quot;&gt;
&lt;%=&quot;Hello, World!&quot;%&gt;
&lt;/syntaxhighlight&gt;

=== Alef++ ===
&lt;syntaxhighlight lang=&quot;text&quot;&gt;
sub say : void {
	System-&gt;out-&gt;println[ $0#0 ];
}

main{
     say[Hello, world!];
}
&lt;/syntaxhighlight&gt;

=== [[Topic:Assembly language|Assembly]] ===

''x86 compatible'' for [[Wikipedia:MS-DOS|MS-DOS]].
&lt;syntaxhighlight lang=&quot;asm&quot; line=&quot;1&quot; start=&quot;1&quot;&gt;
 title Hello World Program 
 dosseg 
 .model small 
 .stack 100h 
 .data 
 hello_message db 'Hello, world!',0dh,0ah,'$' 
 .code
  main  proc
     mov    ax,@data
     mov    ds,ax
     mov    ah,9
     mov    dx,offset hello_message
     int    21h
     mov    ax,4C00h
     int    21h
 main  endp
 end   main
&lt;/syntaxhighlight&gt;

=== BASH ===
&lt;syntaxhighlight lang=&quot;bash&quot;&gt;#!/bin/bash
echo &quot;Hello, world!&quot;&lt;/syntaxhighlight&gt;

=== BASIC ===

==== Applesoft BASIC ====

''Used on Apple ][ machines (Apple ][+, ][e, //c, ][GS)''

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 PRINT &quot;HELLO, WORLD!&quot;
&lt;/syntaxhighlight&gt;

-or-

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 ? &quot;HELLO, WORLD!&quot;
&lt;/syntaxhighlight&gt;

==== Bally/Astrocade Basic ====

''As used on the Bally and Astrocade game systems ca. 1978''

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 PRINT &quot;HELLO, WORLD!&quot;
&lt;/syntaxhighlight&gt;

==== Commodore BASIC ====

''As used on a Commodore 64, ca. 1984''

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 ? &quot;Hello, world!&quot;
&lt;/syntaxhighlight&gt;

====  Dark Basic  ====

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
PRINT &quot;Hello, world!&quot;
&lt;/syntaxhighlight&gt;

==== FreeBASIC and QuickBASIC ====
&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
PRINT &quot;Hello, world!&quot;
SLEEP
&lt;/syntaxhighlight&gt;
 
or:

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
? &quot;Hello, world!&quot;
sleep
&lt;/syntaxhighlight&gt;

==== Intellivision Basic ====

''As used on a Mattel Intellivision, ca. 1983''

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 PRINT &quot;HELLO, WORLD!&quot;
&lt;/syntaxhighlight&gt;

==== Intellivision ECS Basic ====

''As used in the Mattel Intellivision ECS''

&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
10 PRIN &quot;HELLO, WORLD.&quot;
&lt;/syntaxhighlight&gt;

''! not on ECS keyboard. Only 4 char. commands in ECS Basic''

==== Liberty BASIC ====
&lt;syntaxhighlight lang=&quot;qbasic&quot;&gt;
print &quot;Hello, world!&quot;
&lt;/syntaxhighlight&gt;

=== Batch ===
&lt;syntaxhighlight lang=&quot;bash&quot;&gt;
echo Hello, world!
&lt;/syntaxhighlight&gt;

=== [[C]] ===

&lt;syntaxhighlight lang=&quot;c&quot;&gt;
#include &lt;stdio.h&gt;

int main(void) 
{
  printf( &quot;Hello, world!\n&quot; );
  return 0;
}
&lt;/syntaxhighlight&gt;

=== [[Topic:C Sharp|C#]] ===
&lt;syntaxhighlight lang=&quot;csharp&quot;&gt;
using System;

namespace HelloWorld
{
  class Program 
  {
    static void Main() 
    {
      Console.WriteLine(&quot;Hello, world!&quot;);
    }
  }
}
&lt;/syntaxhighlight&gt;

=== [[C++]] ===

&lt;syntaxhighlight lang=&quot;cpp&quot;&gt;
#include &lt;iostream&gt;
using namespace std;

int main()
{
  cout &lt;&lt; &quot;Hello, world!\n&quot;;
  return 0;
}
&lt;/syntaxhighlight&gt;

=== COBOL ===

&lt;syntaxhighlight lang=&quot;cobol&quot;&gt;
      IDENTIFICATION DIVISION.
      PROGRAM-ID. HELLO-WORLD.
      PROCEDURE DIVISION.
          DISPLAY 'Hello, world'.
          STOP RUN.
&lt;/syntaxhighlight&gt;

=== Common Lisp ===
&lt;syntaxhighlight lang=&quot;lisp&quot;&gt;
(print &quot;Hello, world!&quot;)
&lt;/syntaxhighlight&gt;

Or:
&lt;syntaxhighlight lang=&quot;lisp&quot;&gt;
(format t &quot;Hello, world!~%&quot;)
&lt;/syntaxhighlight&gt;

=== [[Delphi]] ===
&lt;syntaxhighlight lang=&quot;delphi&quot;&gt;
begin
  Writeln('Hello, world!');
end.
&lt;/syntaxhighlight&gt;

=== Eztrieve (IBM Mainframe programming language). ===
&lt;code&gt;&lt;pre&gt;JOB NULL

DISPLAY &quot;HELLO, WORLD&quot;

STOP&lt;/pre&gt;&lt;/code&gt;

=== [[Forth]] ===
&lt;code&gt;&lt;pre&gt;: HELLO    .&quot; Hello, world!&quot;  ;
HELLO&lt;/pre&gt;&lt;/code&gt;

=== [[Fortran]] ===
&lt;syntaxhighlight lang=&quot;fortran&quot;&gt;
    PROGRAM HELLO
    PRINT *,'Hello, world'
    STOP
    END
&lt;/syntaxhighlight&gt;

=== [[Go]] ===
&lt;syntaxhighlight lang=&quot;go&quot;&gt;
package main

import &quot;fmt&quot;

func main() {
	fmt.Println(&quot;Hello, World&quot;)
}
&lt;/syntaxhighlight&gt;

=== Haskell ===
&lt;code&gt;&lt;pre&gt;main :: IO ()
main = putStrLn &quot;Hello, world!&quot;&lt;/pre&gt;&lt;/code&gt;

=== [[HTML|Html]] ===
&lt;syntaxhighlight lang=&quot;html4strict&quot;&gt;
&lt;html&gt;
&lt;head&gt;
&lt;title&gt;Hello, world!&lt;/title&gt;
&lt;/head&gt;
&lt;body&gt;
&lt;p&gt;
Hello, world!
&lt;/p&gt;
&lt;/body&gt;
&lt;/html&gt;
&lt;/syntaxhighlight&gt;
=== [[Java]] ===
&lt;syntaxhighlight lang=&quot;java&quot;&gt;
class HelloWorldApp {
    public static void main(String[] args) {
        System.out.println(&quot;Hello World!&quot;); // Display the string.
    }
}
&lt;/syntaxhighlight&gt;
=== [[Portal:JavaScript|JavaScript]] (aka JScript, ECMAScript, LiveScript) ===

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
document.println(&quot;Hello, world!&quot;);
&lt;/syntaxhighlight&gt;

or 

&lt;syntaxhighlight lang=&quot;java&quot;&gt;
alert(&quot;Hello, world!&quot;);
&lt;/syntaxhighlight&gt;

or 


&lt;syntaxhighlight lang=&quot;java&quot;&gt;
document.writeln(&quot;Hello, world!&quot;);
&lt;/syntaxhighlight&gt;

=== [[Luka]] ===
&lt;pre&gt;
print &quot;Hello, world&quot;
&lt;/pre&gt;

or, with proper syntax

&lt;pre&gt;
print( &quot;Hello, world!&quot; );
&lt;/pre&gt;

=== [[w:Oberon (programming language)|Oberon]] ===
&lt;pre&gt;MODULE Hello;

   IMPORT Out;

   PROCEDURE World*;
   BEGIN
      Out.Open;
      Out.String(&quot;Hello, world!&quot;); Out.Ln;
   END World;

END Hello.&lt;/pre&gt;

=== [[OCaml]] ===
&lt;syntaxhighlight lang=&quot;ocaml&quot;&gt;
print_endline &quot;Hello, world!&quot;
&lt;/syntaxhighlight&gt;

=== [[Pascal]] ===
&lt;syntaxhighlight lang=&quot;pascal&quot;&gt;
program HelloWorld;
begin
  writeln( 'Hello, world!' );
end.
&lt;/syntaxhighlight&gt;


=== [[Portal:Perl|Perl]] ===
&lt;syntaxhighlight lang=&quot;perl&quot;&gt;
#!/usr/bin/perl
print &quot;Hello, world!\n&quot;;
&lt;/syntaxhighlight&gt;

=== [[Portal:PHP|PHP]] ===
&lt;syntaxhighlight lang=&quot;php&quot;&gt;
&lt;?php
echo &quot;Hello, world!&quot;;
?&gt;
&lt;/syntaxhighlight&gt;

or (with short_tags enabled in php.ini)

&lt;syntaxhighlight lang=&quot;php&quot;&gt;
&lt;? echo &quot;Hello, world!&quot;; ?&gt;
&lt;/syntaxhighlight&gt;

or (with asp_tags enabled in php.ini)

&lt;syntaxhighlight lang=&quot;php&quot;&gt;
&lt;% echo &quot;Hello, world!&quot;; %&gt;
&lt;/syntaxhighlight&gt;

or 

&lt;syntaxhighlight lang=&quot;php&quot;&gt;
&lt;?=&quot;Hello, world!&quot;?&gt;
&lt;/syntaxhighlight&gt;

=== [[Topic:Python|Python]] ===

With Python 2
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
#!/usr/bin/env python
print 'Hello, world!'
&lt;/syntaxhighlight&gt;
Or with Python 3
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
print(&quot;Hello, world!&quot;)
&lt;/syntaxhighlight&gt;
The first line is used on Unix systems only, and is optional even there. The advantage is that it allows the file to be invoked directly (if &lt;code&gt;chmod +x&lt;/code&gt;), without explicitly specifying the &lt;code&gt;python&lt;/code&gt; interpreter.

=== [[Ruby]] ===
&lt;syntaxhighlight lang=&quot;ruby&quot;&gt;
puts 'Hello, world!'
&lt;/syntaxhighlight&gt;

Another way to do it, albeit more obscure:
&lt;syntaxhighlight lang=&quot;ruby&quot;&gt;
#!/usr/local/bin/ruby
puts 1767707668033969.to_s(36)
&lt;/syntaxhighlight&gt;

=== [[Tcl]] ===
&lt;syntaxhighlight lang=&quot;tcl&quot;&gt;
#!/usr/bin/tclsh
puts &quot;Hello, world!&quot;
&lt;/syntaxhighlight&gt;

=== [[Trekkie]] ===
&lt;pre&gt;
&quot;Computer?&quot;
*Bee bee boo
&quot;Create program 'Hello, World! Picard-alpha-1'&quot;
*Boo boo bee
&quot;Parameters: Display the phrase 'Hello, world!' on the screen the program is executed from until the 
program is terminated.&quot;
*Bee bee
&quot;Save program.&quot;
*Boo bee boo
&lt;/pre&gt;

=== [[Turing]] ===
&lt;pre&gt;
put &quot;Hello World!&quot;
&lt;/pre&gt;

=== [[Visual Basic|Visual Basic 6]] ===
&lt;syntaxhighlight lang=&quot;vb&quot;&gt;
Sub Form1_Load()
    MsgBox &quot;Hello, world!&quot;
End Sub
&lt;/syntaxhighlight&gt;

&lt;ref&gt;is your purpose?
&lt;/ref&gt;== Assignment ==

Create a '''Hello, world!''' program in a language not listed above, then edit this page and add it to the collection.

=== Visual Basic .NET ===

&lt;syntaxhighlight lang=&quot;vbnet&quot;&gt;
Module Module1
 
    Sub Main()
        Console.WriteLine(&quot;Hello, world!&quot;)
    End Sub
 
End Module
&lt;/syntaxhighlight&gt;

=== C ===
Because the tradition of using the phrase &quot;[[w:&quot;Hello,_World!&quot;_program|Hello, world]]!&quot; as a test message was influenced by an example program in the seminal book ''[[w:The C Programming Language (book)|The C Programming Language]]''.&lt;ref&gt;{{cite book | last = Kernighan | first = Brian W. | authorlink = w:Brian W. Kernighan |author2=w:Ritchie, Dennis M. | title = The C Programming Language | edition = 1st | publisher = [[Prentice Hall]] | date = 1978 | location = [[Englewood Cliffs, NJ]] | isbn = 0-13-110163-3 | authorlink2 = Dennis M. Ritchie }}&lt;/ref&gt; that original example is reproduced here.
&lt;syntaxhighlight lang=&quot;text&quot;&gt;
#include &lt;stdio.h&gt;

main( )
{
        printf(&quot;hello, world\n&quot;);
}
&lt;/syntaxhighlight&gt;

=== '''LOLCODE''' ===

&lt;syntaxhighlight lang=&quot;text&quot;&gt;
 HAI
 CAN HAS STDIO?
 VISIBLE &quot;Hello world!&quot;
 KTHXBYE
&lt;/syntaxhighlight&gt;

=== '''Natural ''' ===
&lt;pre&gt;
WRITE 'Hello, world!'
END
&lt;/pre&gt;
&lt;big&gt;Hello, world!&lt;/big&gt;

&lt;ref&gt;即時新聞報導
.&lt;/ref&gt;=== '''XML''' ===

&lt;syntaxhighlight lang=&quot;xml&quot;&gt;
&lt;?xml version=&quot;1.0&quot;?&gt;
&lt;hello&gt;
&lt;messagename=&quot;Hello&quot; /&gt;
&lt;message&gt;
Hello, World!
&lt;/message&gt;
&lt;/hello&gt;
&lt;/syntaxhighlight&gt;

Or with attributes:

&lt;syntaxhighlight lang=&quot;xml&quot;&gt;
&lt;?xml version=&quot;1.0&quot;?&gt;
&lt;hello messagename=&quot;Hello, World!&quot;&gt;
Hello, world!
&lt;/hello&gt;
&lt;/syntaxhighlight&gt;

=== '''[[w:MACRO-11|MACRO-11]]''' ===

&lt;syntaxhighlight lang=&quot;text&quot;&gt;
        .TITLE  HELLO WORLD
        .MCALL  .TTYOUT,.EXIT
HELLO:: MOV     #MSG,R1 ;STARTING ADDRESS OF STRING
1$:     MOVB    (R1)+,R0 ;FETCH NEXT CHARACTER
        BEQ     DONE    ;IF ZERO, EXIT LOOP
        .TTYOUT         ;OTHERWISE PRINT IT
        BR      1$      ;REPEAT LOOP
DONE:   .EXIT

MSG:    .ASCIZ /Hello, world!/
        .END    HELLO
&lt;/syntaxhighlight&gt;
&lt;ref&gt;is your purpose?
&lt;/ref&gt;== Assignment ==

Create a '''Hello, world!''' program in a language not listed above, then edit this page and add it to the collection.

=== Visual Basic .NET ===

&lt;syntaxhighlight lang=&quot;vbnet&quot;&gt;
Module Module1
 
    Sub Main()
        Console.WriteLine(&quot;Hello, world!&quot;)
    End Sub
 
End Module
&lt;/syntaxhighlight&gt;

=== C ===
Because the tradition of using the phrase &quot;[[w:&quot;Hello,_World!&quot;_program|Hello, world]]!&quot; as a test message was influenced by an example program in the seminal book ''[[w:The C Programming Language (book)|The C Programming Language]]''.&lt;ref&gt;{{cite book | last = Kernighan | first = Brian W. | authorlink = w:Brian W. Kernighan |author2=w:Ritchie, Dennis M. | title = The C Programming Language | edition = 1st | publisher = [[Prentice Hall]] | date = 1978 | location = [[Englewood Cliffs, NJ]] | isbn = 0-13-110163-3 | authorlink2 = Dennis M. Ritchie }}&lt;/ref&gt; that original example is reproduced here.
&lt;syntaxhighlight lang=&quot;text&quot;&gt;
#include &lt;stdio.h&gt;

main( )
{
        printf(&quot;hello, world\n&quot;);
}
&lt;/syntaxhighlight&gt;

=== '''LOLCODE''' ===

&lt;syntaxhighlight lang=&quot;text&quot;&gt;
 HAI
 CAN HAS STDIO?
 VISIBLE &quot;Hello world!&quot;
 KTHXBYE
&lt;/syntaxhighlight&gt;

=== '''Natural ''' ===
&lt;pre&gt;
WRITE 'Hello, world!'
END
&lt;/pre&gt;
&lt;big&gt;Hello, world!&lt;/big&gt;

=== '''XML''' ===

&lt;syntaxhighlight lang=&quot;xml&quot;&gt;
&lt;?xml version=&quot;1.0&quot;?&gt;
&lt;hello&gt;
&lt;messagename=&quot;Hello&quot; /&gt;
&lt;message&gt;
Hello, World!
&lt;/message&gt;
&lt;/hello&gt;
&lt;/syntaxhighlight&gt;

Or with attributes:

&lt;syntaxhighlight lang=&quot;xml&quot;&gt;
&lt;?xml version=&quot;1.0&quot;?&gt;
&lt;hello messagename=&quot;Hello, World!&quot;&gt;
Hello, world!
&lt;/hello&gt;
&lt;/syntaxhighlight&gt;

=== '''[[w:MACRO-11|MACRO-11]]''' ===

&lt;syntaxhighlight lang=&quot;text&quot;&gt;
        .TITLE  HELLO WORLD
        .MCALL  .TTYOUT,.EXIT
HELLO:: MOV     #MSG,R1 ;STARTING ADDRESS OF STRING
1$:     MOVB    (R1)+,R0 ;FETCH NEXT CHARACTER
        BEQ     DONE    ;IF ZERO, EXIT LOOP
        .TTYOUT         ;OTHERWISE PRINT IT
        BR      1$      ;REPEAT LOOP
DONE:   .EXIT

MSG:    .ASCIZ /Hello, world!/
        .END    HELLO
&lt;/syntaxhighlight&gt;

== More about Computer Programming ==
*[[Portal:Computer programming|Topic:Computer Programming]]
ca-app-pub-9770990905745158~2653498701

==See also==
{{wikipedia2|Hello world program}}{{commonscat|Hello World}}
* [https://web.archive.org/web/20150404011657/http://en.wikipedia.org/wiki/Hello_world_program_examples Hello world program examples] from Wikipedia (archived copy)

==External links==
* [http://helloworldcollection.de The Hello World Collection] with 500+ Hello World programs

[[Category:Computer programming]]
[[Category:Programming languages]]

[[cs:Hello world!]]
&lt;references /&gt;
[[Category:Change currency]]</text>
      <sha1>e2efveprauzn7dh07skm17qp8bbgomn</sha1>
    </revision>
  </page>
  <page>
    <title>Spanish 2/Chapter 14 (Picnics)</title>
    <ns>0</ns>
    <id>82192</id>
    <revision>
      <id>2805789</id>
      <parentid>2621416</parentid>
      <timestamp>2026-04-21T15:10:55Z</timestamp>
      <contributor>
        <username>~2026-12237-16</username>
        <id>3054184</id>
      </contributor>
      <comment>/* Tastes */ correction of agrio</comment>
      <origin>2805789</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3468" sha1="7acuflluxlvbd2qzd95kg8cczexgwbe" xml:space="preserve">==Chapter 14 (Picnics)==
===Outdoors===
*'''al aire libre''' - outdoors
*'''cielo''' - sky
*'''dar una caminata''' - to take a walk
*'''dentro de''' - inside
*'''fuera''' - outside
*'''hormiga''' - ant
*'''mosca''' - fly
*'''nube''' - cloud
*'''piedra''' - rock
*'''sendero''' - trail
*'''suelo''' - ground, trail

===Eating outdoors===
*'''fogata''' - bonfire
*'''fósforo''' or '''cerilla''' - match
*'''hacer una parrillada''' - to have a barbecue
*'''leña''' - firewood
*'''a la parrilla''' - on the grill
*'''puesto''' - stand

===Foods===
*'''aguacate''' - avocado
*'''asado(a)''' - grilled
*'''asar''' - to grill
*'''bistec''' - steak
*'''carne de res''' - beef
*'''cereza''' - cherry
*'''cesta''' - basket
*'''chuleta de cerdo''' - pork chop
*'''durazno''' - peach
*'''frijoles''' - beans
*'''harina''' - flour
*'''maíz''' - corn
*'''mayonesa''' - mayonnaise
*'''melón''' - melon
*'''mostaza''' - mustard
*'''olor''' - smell, odor
*'''pavo''' - turkey
*'''piña''' - pineapple
*'''sabor''' - taste
*'''salsa de tomate''' - ketchup
*'''sandía''' - watermelon

===Describe foods and the outdoors===
*'''dulce''' - sweet
*'''grasoso(a)''' - fatty
*'''mojado(a)''' - wet
*'''picante''' - spicy
*'''seco(a)''' - dry

===Other words===
*'''acompañar''' - to accompany

===Using ''usted'' and ''ustedes'' commands===
To form an ''Ud.'' or ''Uds.'' command, drop the ''-o'' of the present-tense ''yo'' form and add ''-e'' and ''-en'' for ''-ar'' verbs and ''-a'' and ''-an'' for ''-er'' and ''-ir'' verbs.&lt;br&gt;

====Regular ''Ud.'' and ''Uds.'' commands====
*preparar: '''prepare(n)''' - prepare
*comer: '''coma(n)''' - eat
*servir: '''sirva(n)''' - serve

====Irregular ''Ud.'' and ''Uds.'' commands====
*dar: '''dé, den''' - give
*estar: '''esté, estén''' - be
*poner: '''ponga, pongan''' - put
*ir: '''vaya, vayan''' - go
*ser: '''sea, sean''' - be
*tener: '''tenga, tengan''' - have
*traer: '''traiga, traigan''' - bring

===Using ''por'' in sentences===
*To indicate length of time or distance
*To indicate movement through, along, or around
*To indicate an exchange of one thing for another
*To indicate reason or motive
*To indicate a substitution or action on someone's behalf
*To indicate means of communication or transportation

===Vocabulario adicional===
====Foods====
*'''aceite de oliva''' - olive oil
*'''ají''' - pepper, a hot sauce made with this pepper
*'''apio''' - celery
*'''calamares''' - squid
*'''chorizo''' - sausage
*'''ciruela''' - plum
*'''cordero''' - lamb
*'''espárragos''' - asparagus
*'''espinacas''' - spinach
*'''fruta de estación''' - seasonal fruit
*'''hígado''' - liver
*'''langosta''' - lobster
*'''pepino''' - cucumber
*'''ternera''' - veal
*'''toronja''' - grapefruit
====Tastes====
*'''agrio(a)''' - sour
*'''amargo(a)''' - bitter
*'''cocido(a)''' - cooked
*'''crudo(a)''' - raw
*'''jugoso(a)''' - juicy
*'''salado(a)''' - salty

===Cultural insight (Comida al aire libre)===
[[File:StreetfoodNY.jpg|120px|thumb|Un puesto se vende perritos calientes en Nueva York.]]
Throughout the Spanish-speaking world vendors and food stands make it clear: outdoor eating is very popular among the people. From tamales in Mexico to churros in Spain to fresh fruit anywhere eating outdoors is a favorite. Some dishes are also served with main dishes, such as rice and beans. Eating outdoors is popular among locals because the food is cheap, delicious, and usually prepared faster than at restaurants.


[[Category: Spanish Two]]</text>
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    </revision>
  </page>
  <page>
    <title>Physics (A-level)/Modelling the universe</title>
    <ns>0</ns>
    <id>130039</id>
    <revision>
      <id>2805803</id>
      <parentid>1597351</parentid>
      <timestamp>2026-04-21T18:02:43Z</timestamp>
      <contributor>
        <username>~2026-24466-44</username>
        <id>3067798</id>
      </contributor>
      <comment>/* Redshift */</comment>
      <origin>2805803</origin>
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==Measures of distance==
There several measures of distance that are used when describing the universe. These are used because the SI unit, meter, is inefficient when talking about such large distances. You will need to understand how each distance is determined.
====Astronomical Unit (AU)====
An astronomical unit is the average distance between the Earth and the Sun. It measures to approximately 1.5x10&lt;sup&gt;11&lt;/sup&gt;m
This unit is not often used when discussing any interstellar objects due to it's relatively small magnitude.
====Light-year (ly)====
A light-year is the distance traveled by an electromagnetic wave through free space in one year. It measures to approximately 1x10&lt;sup&gt;16&lt;/sup&gt;m.
&lt;br&gt;
It can be worked out based from the equation '''distance=speed*time''', using the values of the speed of light in a vacuum and the amount of seconds in a year.

====Parsec (pc)====
A parsec is the distance from from which 1AU would subtend to [[w:Minute of arc|1 second of arc]]. It measures to approximately 3.3 ly.
==Structure of the Universe==
===Formation of a star===
You will need to know the steps that happen in the formation of a star
*All stars start as a cloud of dust and gas consisting of mostly hydrogen. This is called a '''nebula'''. Over time the force of gravity attracts these particles together to form progressively larger chunks. These chunks gain a stronger gravitational field strength and attract more matter to itself.
*As this process continues it eventually forms a '''protostar''' which is the very early beginnings of a star and continues to pull matter towards it.
*As more matter is attracted to the protostar and joins onto it, the matter loses gravitational E&lt;sub&gt;p&lt;/sub&gt; which causes a rise in temperature. This in turn will initiate fusion of the hydrogen atoms, releasing energy.
*The fusion reaction sends energy outwards which prevents further gravitational collapse. At this point, it is said to now be a '''Main Sequence Star'''

===Probable evolution of a star===
There are multiple possible fates of a star, generally depending on the star's mass.
&lt;br&gt;&lt;br&gt;
For a star the size of our sun:
*Once all the hydrogen has been used up in the fusion reaction at the star's core, fusion stops. This causes the core to collapse due to the gravitational forces.
*This transfer of E&lt;sub&gt;p&lt;/sub&gt; heats up the hydrogen on outer layers causing it to fuse to helium.
*The outer part of the star expands which also cools it to create a '''Red Giant'''
*When the core reaches about 10&lt;sup&gt;8&lt;/sup&gt;K, helium starts to fuse to create carbon, releasing energy.
*Eventually the outer layers drift away from the stars core and it is left as a dense '''White Dwarf'''. The core of this white dwarf is so dense that matter becomes [[w:Degenerate matter|degenerate]]. At this point fusion has stopped and the white dwarf is left to cool for ever.
&lt;br&gt;
For a star much larger than our sun:
*The temperature and pressure in the core is high enough to fuse hydrogen via the CNO cycle, which is a fusion process in which carbon, nitrogen and oxygen take part to form helium.
*A more massive star will use up all it's hydrogen faster and so will have a shorter lifespan.
*If the star is around 10 times the size of our sun, it will become a '''Super Red Giant''' once the hydrogen has been used up. The core is now hot and dense enough to start the fusion of carbon.
*Once the carbon is used up, the core collapses and starts the next fusion process. This continues until the core consists entirely of iron
*After this, the core will collapse again and cause a huge explosion called a '''supernova'''. All of the material surrounding the core is shot away by the huge shock wave. At this point, energy levels are so high that neutrons combine with other nuclei to form varying elements, some larger than iron
*After the supernova, a small core remains. This is called a '''Neutron Star'''. A neutron star has a density similar to nuclear matter.
*If the star is more than twice the mass of our sun, then a black hole will be produced. This happens when the density is so great that light cannot escape it's gravitational pull.

===Olber's paradox===
Olber's paradox discusses that if the universe is of infinite size and static, then the sky would be completely white with light. This would be due to an infinite amount of stars in every direction.

===Redshift===
Redshift is a process that affects the perceived frequency of a wave as it travels closer or further from an or viewer; it is primarily caused by the doppler effect, where an approaching object appears to increase the frequency of waves emanating from it, and therefore lowering the wavelength of the light which is detected (resulting in Blueshift light) . Similarly, the inverse is observed when an object is moving away from the viewer (known as Redshift light). The concept of redshift is often used within astrophysics to calculate the velocities of galaxies relative to our own, allowing for calculations of how fast they are moving towards or away from us by measuring how 'shifted' the light is. Redshift may also be caused by gravitational forces, or by the expansion of the universe. 

===Hubble's law===

{{CourseCat}}</text>
      <sha1>slw9x1gnprajpznp6tv2laucsfzy8zb</sha1>
    </revision>
    <revision>
      <id>2805805</id>
      <parentid>2805803</parentid>
      <timestamp>2026-04-21T18:06:31Z</timestamp>
      <contributor>
        <username>~2026-24466-44</username>
        <id>3067798</id>
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==Measures of distance==
There several measures of distance that are used when describing the universe. These are used because the SI unit, meter, is inefficient when talking about such large distances. You will need to understand how each distance is determined.
====Astronomical Unit (AU)====
An astronomical unit is the average distance between the Earth and the Sun. It measures to approximately 1.5x10&lt;sup&gt;11&lt;/sup&gt;m
This unit is not often used when discussing any interstellar objects due to it's relatively small magnitude.
====Light-year (ly)====
A light-year is the distance traveled by an electromagnetic wave through free space in one year. It measures to approximately 1x10&lt;sup&gt;16&lt;/sup&gt;m.
&lt;br&gt;
It can be worked out based from the equation '''distance=speed*time''', using the values of the speed of light in a vacuum and the amount of seconds in a year.

====Parsec (pc)====
A parsec is the distance from from which 1AU would subtend to [[w:Minute of arc|1 second of arc]]. It measures to approximately 3.3 ly.
==Structure of the Universe==
===Formation of a star===
You will need to know the steps that happen in the formation of a star
*All stars start as a cloud of dust and gas consisting of mostly hydrogen. This is called a '''nebula'''. Over time the force of gravity attracts these particles together to form progressively larger chunks. These chunks gain a stronger gravitational field strength and attract more matter to itself.
*As this process continues it eventually forms a '''protostar''' which is the very early beginnings of a star and continues to pull matter towards it.
*As more matter is attracted to the protostar and joins onto it, the matter loses gravitational E&lt;sub&gt;p&lt;/sub&gt; which causes a rise in temperature. This in turn will initiate fusion of the hydrogen atoms, releasing energy.
*The fusion reaction sends energy outwards which prevents further gravitational collapse. At this point, it is said to now be a '''Main Sequence Star'''

===Probable evolution of a star===
There are multiple possible fates of a star, generally depending on the star's mass.
&lt;br&gt;&lt;br&gt;
For a star the size of our sun:
*Once all the hydrogen has been used up in the fusion reaction at the star's core, fusion stops. This causes the core to collapse due to the gravitational forces.
*This transfer of E&lt;sub&gt;p&lt;/sub&gt; heats up the hydrogen on outer layers causing it to fuse to helium.
*The outer part of the star expands which also cools it to create a '''Red Giant'''
*When the core reaches about 10&lt;sup&gt;8&lt;/sup&gt;K, helium starts to fuse to create carbon, releasing energy.
*Eventually the outer layers drift away from the stars core and it is left as a dense '''White Dwarf'''. The core of this white dwarf is so dense that matter becomes [[w:Degenerate matter|degenerate]]. At this point fusion has stopped and the white dwarf is left to cool for ever.
&lt;br&gt;
For a star much larger than our sun:
*The temperature and pressure in the core is high enough to fuse hydrogen via the CNO cycle, which is a fusion process in which carbon, nitrogen and oxygen take part to form helium.
*A more massive star will use up all it's hydrogen faster and so will have a shorter lifespan.
*If the star is around 10 times the size of our sun, it will become a '''Super Red Giant''' once the hydrogen has been used up. The core is now hot and dense enough to start the fusion of carbon.
*Once the carbon is used up, the core collapses and starts the next fusion process. This continues until the core consists entirely of iron
*After this, the core will collapse again and cause a huge explosion called a '''supernova'''. All of the material surrounding the core is shot away by the huge shock wave. At this point, energy levels are so high that neutrons combine with other nuclei to form varying elements, some larger than iron
*After the supernova, a small core remains. This is called a '''Neutron Star'''. A neutron star has a density similar to nuclear matter.
*If the star is more than twice the mass of our sun, then a black hole will be produced. This happens when the density is so great that light cannot escape it's gravitational pull.

===Olber's paradox===
Olber's paradox discusses that if the universe is of infinite size and static, then the sky would be completely white with light. This would be due to an infinite amount of stars in every direction.

===Redshift===
Redshift is a process that affects the perceived frequency of a wave as it travels closer or further from an or viewer; it is primarily caused by the [[wikipedia:Doppler_effect|Doppler effect]], where an approaching object appears to increase the frequency of waves emanating from it, and therefore lowering the wavelength of the light which is detected (resulting in Blueshift light) . Similarly, the inverse is observed when an object is moving away from the viewer (known as Redshift light). The concept of redshift is often used within astrophysics to calculate the velocities of galaxies relative to our own, allowing for calculations of how fast they are moving towards or away from us by measuring how 'shifted' the light is. Redshift may also be caused by gravitational forces, or by the expansion of the universe. 

===Hubble's law===

{{CourseCat}}</text>
      <sha1>8gp2mga3t7pncu9uk4x9xp01aovl8y1</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2805761</id>
      <parentid>2805639</parentid>
      <timestamp>2026-04-21T13:34:19Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2805761</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3135" sha1="8ehdu5tczx4m24ilxp08pz8of7qv8gz" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260421.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>8ehdu5tczx4m24ilxp08pz8of7qv8gz</sha1>
    </revision>
    <revision>
      <id>2805902</id>
      <parentid>2805761</parentid>
      <timestamp>2026-04-22T07:05:17Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2805902</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3135" sha1="gr5s5a9emeqe222ea6ndkpcpom4523k" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260422.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>gr5s5a9emeqe222ea6ndkpcpom4523k</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2805775</id>
      <parentid>2805643</parentid>
      <timestamp>2026-04-21T13:58:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2805775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="f4o57k9bigyy427eis0twi5a1fkewlb" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

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

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

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

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

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

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

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

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

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

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

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

====Geometric Series Examples====

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

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

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

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

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

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260421.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

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

* Double Pole Case

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

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

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

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

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

[[Category:Complex analysis]]</text>
      <sha1>f4o57k9bigyy427eis0twi5a1fkewlb</sha1>
    </revision>
    <revision>
      <id>2805911</id>
      <parentid>2805775</parentid>
      <timestamp>2026-04-22T07:21:41Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2805911</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="j4cyh8n82yiaki6ugfvdh39b8bxnoho" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

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

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

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

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

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

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

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

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

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

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

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

====Geometric Series Examples====

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

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

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

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

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

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260422.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

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

* Double Pole Case

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

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

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

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

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

[[Category:Complex analysis]]</text>
      <sha1>j4cyh8n82yiaki6ugfvdh39b8bxnoho</sha1>
    </revision>
  </page>
  <page>
    <title>User:Mortikaii/sandbox</title>
    <ns>2</ns>
    <id>203461</id>
    <revision>
      <id>2805915</id>
      <parentid>1477948</parentid>
      <timestamp>2026-04-22T07:31:22Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Removing [[:c:File:Fallout_4_logo.png|Fallout_4_logo.png]], it has been deleted from Commons by [[:c:User:Masur|Masur]] because: [[:c:COM:DW|Derivative work]] of non-free content ([[:c:COM:CSD#F3|F3]]).</comment>
      <origin>2805915</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="340" sha1="g0yhslpaxgo32o6zvvkixq61fo30zkm" xml:space="preserve">This is the history of UW-Stout[[File:Panel-77688519-image-17f675ff937bdb09-320.jpeg|alt=welcome pic|thumb|320x320px|welcome]]
&lt;ref&gt;Dan. (2016) Dan[dan] afadfsf&lt;/ref&gt;

== Stout ==
stuffs 
=== STUFF 2 ===

======= qerqwr =======

== heading 2 ==
stuffer

== heading 3 ==
stuffy

== external links ==
[http://www.google.com Google]

== Ref ==</text>
      <sha1>g0yhslpaxgo32o6zvvkixq61fo30zkm</sha1>
    </revision>
  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2805865</id>
      <parentid>2805040</parentid>
      <timestamp>2026-04-22T04:36:53Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2805865</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="b7vqw68zywqo554xrm8s88q4dex5wzs" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260420.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
      <sha1>b7vqw68zywqo554xrm8s88q4dex5wzs</sha1>
    </revision>
    <revision>
      <id>2805867</id>
      <parentid>2805865</parentid>
      <timestamp>2026-04-22T04:38:07Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2805867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="jjh016fqo5rb4wm21wwqp13tfutqv53" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260421.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
      <sha1>jjh016fqo5rb4wm21wwqp13tfutqv53</sha1>
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  <page>
    <title>Real World Lab</title>
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      <timestamp>2026-04-22T10:59:23Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Main Use-Cases of RWL */</comment>
      <origin>2805930</origin>
      <model>wikitext</model>
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      <text bytes="15264" sha1="4a2trw3y8e4qydwy264nc1fj716kvzz" xml:space="preserve">[[File:aframe360image.png|thumb|300px|[https://aframe.io/examples/showcase/sky/ AFrame 360 Image]  - Look around by draging mouse with mouse button presse]]
[[File:Gibraltar 1 3d model, created using photogrammetry.ogg|thumb|right|3D models assigned to the location, where a skull was found - video example of a 3D model - [[Wikipedia:Gibraltar 1|Gibraltar 1]] [[Wikipedia:Neanderthal|Neanderthal]] skull 3-D wireframe model (see [[Photogrammetry]]) ]]

[[File:See im Wald.jpg|thumb|Lake in the forest - as an example of an Real World Lab - exploration of flora and fauna]]
[[File:01 Auckland New Zealand-1000137.jpg|thumb|View single Image - Sky Tower, Auckland, New Zealand]]
[[File:Sky Tower Auckland in AFrame Panorama.jpg|thumb|[https://niebert.github.io/HuginSample Panorama Image with AFrame]]]
This learning resource introduces Wikiversity authors into the concept of Real World Labs and supports the authors in building a Wikiversity Learning Resource for the [[Risk_Literacy/Real_World_Labs|Real World Lab]].

== Definition: Real World Lab ==
A real world lab is learning environment 
* located in the specific area, building, ... which is not the regular classroom location and supports the interaction between the scientific world and civil society,
* encourage the interaction with the objects, the local environment or with processes, which are specific for the location of the Real World Lab,
* learning objects go beyond the exploration of the specific location
* engaging people (students, citizens, visitors, ...) to interact with local and digital objects of the Real World Lab.
* (optional) learning environments in the real world lab (outside classroom, seminar room) are linked to learning environment inside the classroom,
* (optional) digital and non-digital material are collected, documented and analyzed.

== Submodules ==
* [[/Introduction/]]
* [[/Planning/]]
* [[/Open Data/|Open Data for a Learning Resource]]
* [[/3D Model/|3D Model of an area or learning objects]]

== Learning Tasks ==
* Consider the conception of a [https://niebert.github.io/Virtual-Reality-Museum/ virtual museum]&lt;ref&gt;Crecropia (2016) The Hall - Virtual Museum - Demo: https://niebert.github.io/Virtual-Reality-Museum/ GitHub-repository: https://github.com/Cecropia/thehallaframe - Fork: https://github.com/niebert/Virtual-Reality-Museum (accessed 2023/02/01)&lt;/ref&gt; and the exploration of the space. What are the analogies and differences with a real-world laboratory where digital visualizations support the exploration of real space?

== Main Use-Cases of RWL ==
* '''(Pre-Visit)''' Preparation of visit with learning tasks and learning resources, that support the learners in being focussed on the learning objectives.
* '''(During-Visit)''' Support the real visit with additional of the RWL with additional Learning Resources (e.g. Quiz, Mulitmedia from WikiMedia Commons, ... in Wikiversity) 
* '''(Post-Visit)''' support of post-visit with additional retrospective learning tasks and learning resources.
* '''([https://niebert.github.io/aframe360navigation Virtual-Visit]&lt;ref&gt;Aframe 460 Degree Prototype (2021) GitHub repository for the Wikiversity learning resource - URL: https://niebert.github.io/aframe360navigation - Download: [https://github.com/niebert/trifels/archive/main.zip ZIP of Repository] - (accessed 2021/01/15) &lt;/ref&gt;)''' if students are not able to visit a specific area on earth (e.g. due to travel time and travel expenses), then a virtual visit could provide an opportunity explore a Real World Lab at least virtually by visual exploration and learning task, e.g. 
** Fiji Islands and the need for migration due to the rise of sea level and explore how a place looked like 50 years ago and now.
**  provide learning resources and the scientfic evidence for [[w:Global_warming|Global Warming]] and [[w:Greenhouse Gas|Greenhouse Gas]].
** (Loss of Value for Fossile Energy Sources) Analyse the hypothesis: The acceptance of current scientific evidence for climate change and the  incorporation in economical decision making will lead to the loss of value of Fossile Energy Sources, because human-kind had to stop burning those resources (keep in mind, that fossile energy sources are not used for fuel and heating alone).

== 360 Degree Images - VR-AR ==
Explore options to support a visit to a real world lab with 360 degree images, virtual reality, mixed reality, ... 
* '''[https://niebert.github.io/WikiCommons2AFrame/index.html WikiCommons2AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame&quot;/&gt;  Create 360-degree scenes in AFrame as a ZIP for offline (and online) use or create links with a list of 360-degree image used as support material for exploration of a location e.g. for (e.g. [[Real World Lab]]).

== Subtopics ==
* [[Risk_Literacy/Real_World_Labs/web-based_Exploration|webbased tools for Real World Labs]]
* [[3D Modelling]] is used to create digital learning resources that allow the exploration of RWL in a pre, during, post, or virtual setting for a learning process.
* [[Photogrammetry]] to create a 3D model of objects in Real World Lab by a set of images from a different angle

== Image Map Graph ==
Create a visual representation ([[Image Map]] Graph - IMG) in LibreOffice Draw and highlight those areas with links to other web resource. This can also be used for the learners to get an overview what can be explored in a learning resource.

:[[File:Imagemap graph.png|550px|Image Map Graph for Navigation]]

The image abover can also be used in the [https://niebert.github.io/imgmap ImageMap Editor] to link the Image Map Graph to the image maps as nodes of the graph.
* Assign a [[Water|Wikiversity Learning Resource about Water]] to an area in the image map or 
* Assign a [https://niebert.github.io/JSON3D4Aframe/mods3d/water_molecule_aframe.html 3D-Model about the water molecule] generate by [[3D Modelling]] to the small river on the image.

== Examples of Real World Location for Labs ==
The following list enumerates possible example locations for Real World Labs. The visualizations are created as 360&lt;sup&gt;o&lt;/sup&gt; degree images create with [https://niebert.github.io/WikiCommons2AFrame/ WikiCommons2AFrame]&lt;ref&gt;[https://niebert.github.io/WikiCommons2AFrame/ WikiCommons2AFrame] (2018) Engelbert Niehaus - Tool for Wikiversity Learning Environments to create a web-based 360&lt;sup&gt;o&lt;/sup&gt; degree images for an equirectangular image stored in WikiCommons - 360&lt;sup&gt;o&lt;/sup&gt; degree images - URL: https://niebert.github.io/WikiCommons2AFrame/ (accessed 2019/04/10)&lt;/ref&gt;
* '''[https://niebert.github.io/aframe360navigation Geographical Location: River]''', The [https://niebert.github.io/aframe360navigation Aframe example] lets the learners explore the river Rhine close to Cologne.
** Biology: flora fauna around the river.
** Architecture: bridges, flood protection measures,...
** History: archeological site, foundation of building, bridges, role of transportation over the river, the river as border between countries
** Technology: water purification, drinking water
** Economy: the role of water in society, transportation of goods over the river
** Physics: fluid dynamic and transformation of river bed, land-water and water-land interaction
** Mathematics: mathematical modelling of topics mentioned before and calculation of models specific to the geolocation of the real world lab.
* '''(Technology)''' Exploration with a [https://niebert.github.io/WikiCommons2AFrame/wikicommons2aframe.html?domain=weblink&amp;skyimage=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fc%2Fc7%2FDatteln_Schleuse_Panorama.jpg&amp;useaframecode=yes&amp;aframecode=%7B%0D%0A++%22a-text%22%3A+%7B%0D%0A++++%22font%22%3A%22kelsonsans%22+%2C%0D%0A++++%22value%22%3A%22Sluice+in+Datteln+-+Germany%22%2C%0D%0A++++%22color%22%3A%22white%22%2C%0D%0A++++%22width%22%3A%226%22%2C%0D%0A++++%22position%22%3A%22-3.5+2.25+-2.5%22%2C%0D%0A++++%22rotation%22%3A%220+15+0%22%0D%0A++%7D%0D%0A%7D%0D%0A 360 degree image of a sluice in Germany]
* '''Mining Area''' &lt;kbd&gt;[https://commons.wikimedia.org/wiki/File:F%C3%B6rderturm_Zeche_Holland_Panorama.jpg Förderturm_Zeche_Holland_Panorama.jpg]&lt;/kbd&gt; -- [https://niebert.github.io/WikiCommons2AFrame/wikicommons2aframe.html?domain=weblink&amp;skyimage=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fb%2Fbf%2FF%25C3%25B6rderturm_Zeche_Holland_Panorama.jpg&amp;useaframecode=yes&amp;aframecode=%7B%0D%0A++%22a-text%22%3A+%7B%0D%0A++++%22font%22%3A%22kelsonsans%22+%2C%0D%0A++++%22value%22%3A%22Foerderturm+der+Zeche+Holland%22%2C%0D%0A++++%22color%22%3A%22white%22%2C%0D%0A++++%22width%22%3A%226%22%2C%0D%0A++++%22position%22%3A%22-3.5+2.25+-2.5%22%2C%0D%0A++++%22rotation%22%3A%220+15+0%22%0D%0A++%7D%0D%0A%7D%0D%0A++++++++++ Förderturm der Zeche Holland] - link created with [https://niebert.github.io/WikiCommons2AFrame WikiCommons2AFrame] with [https://commons.wikimedia.org/wiki/File:F%C3%B6rderturm_Zeche_Holland_Panorama.jpg equirectangular image from WikiCommons].
* '''[https://github.com/r23/Virtual-Reality-Museum Museum]''' in [[3D Modelling|AFrame]]
* '''Zoo'''
* '''Nature/Environment''' &lt;kbd&gt;[https://niebert.github.io/HuginSample/img/rieselfelder1.jpg  Rieselfelder Muenster, Germany]&lt;/kbd&gt; with https://niebert.github.io/HuginSample/img/rieselfelder1.jpg --  Generated Link for '''[https://niebert.github.io/WikiCommons2AFrame/wikicommons2aframe.html?domain=weblink&amp;skyimage=https%3A%2F%2Fniebert.github.io%2FHuginSample%2Fimg%2Frieselfelder1.jpg&amp;useaframecode=yes&amp;aframecode=%7B%0D%0A++%22a-text%22%3A+%7B%0D%0A++++%22font%22%3A%22kelsonsans%22+%2C%0D%0A++++%22value%22%3A%22Generated+with+WikiCommons2AFrame%22%2C%0D%0A++++%22color%22%3A%22blue%22%2C%0D%0A++++%22width%22%3A%226%22%2C%0D%0A++++%22position%22%3A%22-3.5+2.25+-2.5%22%2C%0D%0A++++%22rotation%22%3A%220+15+0%22%0D%0A++%7D%0D%0A%7D%0D%0A 360&lt;sup&gt;o&lt;/sup&gt;-Image]'''
* '''Farm'''
* '''City'''  &lt;kbd&gt;[https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg  Saumarkt Karlsruhe-Durlach, Germany]&lt;/kbd&gt; with https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg --  Generated Link for '''[https://niebert.github.io/WikiCommons2AFrame/wikicommons2aframe.html?domain=weblink&amp;skyimage=https%3A%2F%2Fniebert.github.io%2FHuginSample%2Fimg%2Fdurlach_saumarkt.jpg&amp;useaframecode=yes&amp;aframecode=%7B%0D%0A++%22a-text%22%3A+%7B%0D%0A++++%22font%22%3A%22kelsonsans%22+%2C%0D%0A++++%22value%22%3A%22Durlach+-+Saurmarkt%22%2C%0D%0A++++%22color%22%3A%22white%22%2C%0D%0A++++%22width%22%3A%226%22%2C%0D%0A++++%22position%22%3A%22-3.5+2.25+-2.5%22%2C%0D%0A++++%22rotation%22%3A%220+15+0%22%0D%0A++%7D%0D%0A%7D%0D%0A  360&lt;sup&gt;o&lt;/sup&gt;-Image]'''
* '''Kindergarten/Playground'''
* '''[http://www.digitales-forum-romanum.de/?lang=en Archeological Site]&lt;ref&gt;Digital Forum Romanum (2019) Humbold University, Berlin, URL: http://www.digitales-forum-romanum.de/?lang=en (accessed 2021/07/02)&lt;/ref&gt;''' learning task is to create e.g. a small temple with [https://niebert.github.io/WikiCommons2AFrame WikiCommons2AFrame] and assign an [[equirectangular]] image to the 3D scene - see &lt;kbd&gt;[https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Aldara_parks.jpg/1280px-Aldara_parks.jpg Aldara_parks.jpg]&lt;/kbd&gt; -- Generated Link for '''[https://niebert.github.io/WikiCommons2AFrame/wikicommons2aframe.html?domain=weblink&amp;skyimage=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fd%2Fd8%2FAldara_parks.jpg&amp;useaframecode=yes&amp;aframecode=%7B%0D%0A++%22a-text%22%3A+%7B%0D%0A++++%22font%22%3A%22kelsonsans%22+%2C%0D%0A++++%22value%22%3A%22Aldara+Park%22%2C%0D%0A++++%22color%22%3A%22white%22%2C%0D%0A++++%22width%22%3A%226%22%2C%0D%0A++++%22position%22%3A%22-3.5+2.25+-2.5%22%2C%0D%0A++++%22rotation%22%3A%220+15+0%22%0D%0A++%7D%0D%0A%7D%0D%0A 360&lt;sup&gt;o&lt;/sup&gt;-Image]'''

== Icon Sets for Learning Resource ==
* Use Icon Sets for your Learning Resource to express, what kind of interaction with the Real World Lab you expect from the learner.

== Learning Tasks ==
* '''([[Risk Literacy/Real World Labs/web-based Exploration|Web-based Exploration]])''' Analyze how a [[Risk Literacy/Real World Labs/web-based Exploration|web-based exploration]] can be used in conjunction with a real visit or preparation of the real visit of the real world lab.
* '''(Markers on a Map)''' Explore the [https://niebert.github.io/Markers4Map/ OpenSource HTML5 Tool] to create markers for your Real World Lab on a map. Use [https://niebert.github.io/Markers4Map/ Markers4Map] and create interesting geolocations on the map and explain e.g. what kind of experiments can be performed at the geolocation ([https://niebert.github.io/openlayer_display_markers/viewicons.html?mapcenter=-1.81185%2C+52.443141&amp;zoom=5&amp;jsondata=%5B%7B+%22geolocation%22+%3A+%5B-0.14467470703124907%2C51.493889053694915%5D%2C%22name%22+%3A+%22%3Cb%3E%3Ca+href%3D%5C%22https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLondon%5C%22+target%3D%5C%22_blank%5C%22%3ELondon%3C%2Fa%3E%3C%2Fb%3E%3Cbr%3ELondon+created+a+Open+Innovation+Ecosystem+for+SDG+Clean+Water+and+Sanitation%22%7D%2C%0D%0A%7B+%22geolocation%22+%3A+%5B-1.81185%2C51.243141%5D%2C%22name%22+%3A+%22%3Cb%3E%3Ca+href%3D%5C%22https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBirmingham%5C%22+target%3D%5C%22_blank%5C%22%3EBrimingham%3C%2Fa%3E%3C%2Fb%3E%3Cbr%3EBirmingham+created+SDG-activities+about+SDG7+Clean+and+affordable+Energy+and+SDG3+Health%22%7D%5D Example Map with two markers]).
* '''(Digital and Non-Digital Aspects of the RWL)''' Create a Real World Lab that allows learners and citzens to interact with the environment. Identify digital areas of exploration that augment the experience with real world environment. 
** '''(Time)''' Explore the visual appearance of the location 
*** '''(Winter)''' in winter by using [[Virtual Reality]], when it is summer and 
*** '''(Summer)''' vice versa a [[Virtual Reality|VR experience]] of the location during summer when it is winter. 
*** ''(History)'' explore visual experience of the location 50 years before (e.g. plants, buildings) or before/after flooding, ... (Disaster)
** '''(Scale)''' Explore visualisation 
*** ''(Microscope)'' under the mircoscope as video of organisms, that can be found in the river,
*** ''([[Satellite Technology|Satellite Image]])'' explore satellite image of the region during extreme event due to [[Climate change|Climate Change]]
*** ''(Drone Video)'' provide a digital bird view or the region.
*** ''(Digital Maps)'' maps used to understand the geographical context of the Real World Lab

== See also ==
* [https://niebert.github.io/WikiCommons2AFrame/ WikiCommons2AFrame] - Open Source webbased tool for Wikiversity learning resources to create links for 360&lt;sup&gt;o&lt;/sup&gt; degree images, that are stored in WikiCommons.
* [[AppLSAC]] for privacy-friendly data collection in Real World Labs, that allow offline data collection with in installed browser as runtime environment for the WebApp.
* [[Risk Literacy/Real World Labs]]
* [[Risk Literacy/Real World Labs/web-based Exploration]]
* [[Image_Map/Tutorial]]
* [[c:Category:3D Model - STL|3D Models STL in WikiCommons]] contains 3D models a file format STL - for integration of 3D models in learning resources in Wikiversity.

== References ==


&lt;noinclude&gt;[[de:Reallabor]]&lt;/noinclude&gt;
[[Category:Real World Lab]]
[[Category:Virtual reality]]
[[Category:3D modeling]]
[[Category:Equirectangular]]</text>
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      <text bytes="6879" sha1="diwqllgxzppy4z3gr1tmrzx5v3c4rpw" xml:space="preserve">[[File:An early concept design of the ERIS instrument.jpg|thumb|Three-dimensional model of a [[Wikipedia:spectrograph|spectrograph]]&lt;ref&gt;ERIS Project Starts - http://www.eso.org/public/announcements/ann13054/ - (accessed 2013/06/14) newspaper ESO Announcement&lt;/ref&gt; ]]

[[File:Gibraltar 1 3d model, created using photogrammetry.ogg|thumb|right|[[Wikipedia:Gibraltar 1|Gibraltar 1]] [[Wikipedia:Neanderthal|Neanderthal]] skull 3-D wireframe model (see [[Photogrammetry]]) ]]

[[File:Steps of forensic facial reconstruction - Virtual Mummy - cogitas3d.gif|right|300px|thumb|Steps of [[Wikipedia:forensic facial reconstruction|forensic facial reconstruction]] of a [[Wikipedia:mummy|mummy]] made in [[Wikipedia:Blender (software)|Blender]] by the Brazilian 3D designer [[:pt:Cícero Moraes|Cícero Moraes]].]]

[[File:Water molecule aframe.png|300px|thumb|[https://niebert.github.io/JSON3D4Aframe/mods3d/water_molecule_aframe.html AFrame Water Molecule] created with [https://niebert.github.io/JSON3D4Aframe/ JSON3D4Aframe] ]]

[[File:Crystal lattice with hugin sky background.png|thumb|Screenshot  of [https://niebert.github.io/HuginSample/crystal_lattice_sky.html Crystal Lattice with Hugin Sky Background in AFrame]. Use cursor keys to move crystal lattice.]]

&lt;!-- Comment in Wikiversity --&gt; 
This learning resource support authors in Wikiversity to provide 3D models in learning modules. Furthermore this module can be used for learners themselves to create web-based 3-D models. The course focuses on web-based three-dimensional models, because they how to be accessible directly by a link in the Wikiversity content. 

== History ==
The learning resouces follows the [[Open Community Approach]] and therefore the used software is Open Source and the course is setup up in Wikiversity, so that the community of learners can learn, modify and extend the learning resource. All tools presented in this course Open source tools, they are freely accessible for learners.

== Learning Objective ==
The course is based on two major steps: See and explore web-based models and create your own three-dimensional models if you decide they are useful for you learning module in Wikiversity.
* '''[[/Examples/|Explore 3D Models]]:''' Learn, what you can do with [[/Examples/|3D models in a browser]].
* '''[[/Create 3D Models/|Create 3D Models]]:''' Learn, how to [[/Create 3D Models/|Create 3D your own models]] with OpenSource tool.
* '''[[/360 Degree Video/]]''' as realistic background video in 3D models
* '''[[/Projective Geometry/]]''' Projective geometry introduces to the workflow from objects in the 3D space into the twodimensional plane.
* '''[[Photogrammetry]]''' addresses the opposite workflow from multiple twodimensional images towards a 3D model.

== Subtopics ==
* [[3D Scan]] - use scanning devices to create 3D Models of an object
* [[Photogrammetry]]&lt;ref&gt;Mikhail, E. M., Bethel, J. S., &amp; McGlone, J. C. (2001). Introduction to modern photogrammetry. New York.&lt;/ref&gt; - Take a snapshot of an object from different angles and generate a 3D Model with software (e.g. [[Photogrammetry/Regard3D|OpenSource Regard3D]])
* [[Perspective_Drawing_on_Mirror|Perspective Drawing on a Mirror]] with [[Geogebra]]
* [[/Geographic Terrains/]]
* [[/Animal Models/]]
* [[/3D Interior Design/]]
* [[/3D Models in Mathematics/]]

== Learning Task ==
* '''([[/Create_3D_Models/|Your first 3D Model of a Cup]])''' Perform the [[/Create_3D_Models/|Blender Cup Tutorial]]!
* '''([[Screencasting]])''' Learn how to [https://www.youtube.com/watch?v=qeDNgLBVPLU create a Screencast]&lt;ref&gt;OBS Tutorial: Open Broadcaster Software - Screencast/Video Screen Capture (OBS Studio/Multiplatform) by TanUv90 (2016) - https://www.youtube.com/watch?v=qeDNgLBVPLU (accessed 2018/01/11)&lt;/ref&gt; with [https://obsproject.com/ Open Broadcasting Software (OBS)]!
* '''(Train the teachers)''' Create your own learning video and share your results on Wikiversity!
* Export the Cup as [https://aframe.io/docs/0.7.0/components/gltf-model.html GLTF model] in blender and place your [[3D_Modelling/Create_3D_Models/AR.js|cup on a marker with AR.js]].
* '''([[Open Educational Resources|3D Models and OER]])''' Analyse the options for 3D-Printing for sharing learning material as [[Open Educational Resources]].
* '''([[Computer-aided design|Computer-Aided Design - CAD]])''' 
** Explain why and how computer-aided design needs a 3D modelling infrastructure. 
** Describe different pathways to create a non-digital objects from a digital 3D model (small objects in 3D printer, print houses, mirco, nano scale, ... - include also cost efficience and required number of mockups and prototypes for the design process). 
** Compare [[transport]] of non-digital products with a transport of digital 3D model over the internet and a 3D print at the location where the product is needed.
** Space agencies plan to land on Mars and pay-load to other planets is extremly costly. Sending a digital 3D model from on planet to another planet and produce the product from the 3D model is an other alternative. Compare those approaches for space traffic management and transfer the setting to applications on earth (e.g. printing required resources in an emergency case in remote areas, where transport of non-digital resources would take much more time.
** Design a passenger cell for [[Intermodal Public Transport]] and create a [[w:en:Blender_(software)|Blender]] animation that shows [[Interoperability]] in a 3D model. Show the process from a idea to an animated 3D model, which shows how a passenger enters a kind of box (passenger cell) and the passenger cell changes its  propulsion unit, e.g. electro vehicle from home, load several passenger cell on a and transports those to a harbor. Aggregate passenger cells on a ship and let the passenger use service infrastructure on that ship. In the destination harbor load  a single passenger cell on a trailer and transport the passengercell finaly by an aeroplane to the final destination. 
*** What are the required modelling elements that you need?
*** You can import and duplicate those elements in a way that modifications of design will affect all uses passenger cells.
*** Plan the animation in Blender with a storyboard.

== Tools for Learning Resource ==
* [[Blender]]
* [[JSON3D4Aframe]]
* [https://niebert.github.io/WikiCommons2Aframe WikiCommons2Aframe]&lt;ref&gt;WikiCommons2Aframe (2018-25) Create Web pages with 360 degree - equirectangular images URL: https://niebert.github.io/WikiCommons2Aframe - GitHub repository https://www.github.com/niebert/WikiCommons2Aframe&lt;/ref&gt;

== See also ==
* [[2D Animation]]
* [[w:en:Blender_(software)|Blender (software)]]
* [[Real World Lab]]
* [https://trackingjs.com TrackingJS] - tracking of objects in a webcam and videos just with [[JavaScript]]
* [[Stereoscopy]]
* [[Photogrammetry]]

== References ==

[[Category:3D modeling]]</text>
      <sha1>diwqllgxzppy4z3gr1tmrzx5v3c4rpw</sha1>
    </revision>
  </page>
  <page>
    <title>3D Modelling/Create 3D Models</title>
    <ns>0</ns>
    <id>229294</id>
    <revision>
      <id>2805926</id>
      <parentid>2620048</parentid>
      <timestamp>2026-04-22T10:51:32Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* 3D Software Packages */</comment>
      <origin>2805926</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8005" sha1="bfiqjt0jhre8pnom197r6lnohvcodcd" xml:space="preserve">[[File:Water molecule aframe.png|300px|thumb|Water Molecule in Aframe - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Blender 2.79.png|thumb|350px|Blender Software - OpenSource - Windows, Linux, MacOSX]]
[[File:Water molecule kanji.png|300px|thumb|Water Molecule with [[Augmented Reality]] AR.js on Kanji Marker - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Sodium-chloride-3D-ionic.png|thumb|Crystal structure of sodium chloride (table salt) [https://niebert.github.io/JSON3D4Aframe/mods3d/cristal_lattice_aframe.html 3D Model in AFrame] created with [https://niebert.github.io/JSON3D4Aframe/ JSON3D4Aframe] ]]

The following learning resource supports you as an author to create you own three-dimensional models (see also demo 3D models provided by Lee Stemkoski&lt;ref&gt;Lee Stemkoski (2021) Aframe Examples on GitHub - Demo URL: https://stemkoski.github.io/A-Frame-Examples/ - GitHub repository: https://github.com/stemkoski/A-Frame-Examples [https://github.com/stemkoski/A-Frame-Examples/archive/refs/heads/master.zip ZIP] (accessed 2024/04/13) &lt;/ref&gt;. Provided tools are open source and can be used either for video animation three-dimensional space (e.g. genetic processes that are not visible in a microscope) are you construct 3-D models there are placed in the camera image of mobile device on specific markers (e.g. in [[3D_Modelling/Examples/|AR.js]]).

== Preparation ==
If you are following the [[Open Community Approach]] the created models can be adapted to local and regional requirements and constraints or specifically to target groups of learners that will use the models. When adaptation of models include specific information about the learner's environment then 3D models cannot be shared beyond the [[Digital Learning Environment]] in which the learner uses the open 3D model. In other use-cases the 3D model in the learning environment it is recommended to share the 3D models with the community of authors and learners on Wikiversity for further improvement (see also [[Open Innovation Ecosystem]]).
* currently 3D models cannot be integrated directly into Wikiversity, so files must be stored and shared on an other WebServer, that supports [[Version control|version control]] during the development process of the 3D models. 
* in this learning resource we use [https://www.github.com GitHub] and take advantage of the [[Version_control|version control system]] to share the files in the &lt;kbd&gt;/docs&lt;/kbd&gt; folder on GitHub. Feel free to use any other for the community available version control system. Version control is helpful to have the choice to revert changes due to drawbacks and challenges with the new version.
** Create a new [https://www.github.com GitHub] account if you do not have on (e.g. &quot;myacount&quot;).
** Login into [https://www.github.com GitHub] with your account.

== 3D Software Packages ==
* '''[[/360-Degree AFrame/]]''' Integrate 360 Images in Aframe for digital exploration of locations.
* '''[[/Wiki_360_Degree_AFrame/|360-Degree Image in Wikiversity]]''' Create a link to embed [[/360-Degree AFrame/|360-Degree Image]] from Wiki Commons in AFrame.
* '''[https://niebert.github.io/WikiCommons2AFrame/index.html WikiCommons2AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame&quot;/&gt; Niehaus, Bert (2026) WikiCommons2Aframe URL: https://niebert.github.io/WikiCommons2AFrame/index.html - Create 360-degree scenes in AFrame as a ZIP for offline (and online) use or create links with a 350-degree image that is online available as for online use. 
* '''[[/Blender/|Blender - Photorealistic 3D Modelling with Physics, Animation, ...]]:''' (e.g. Fluid Dynamics and Wind)&lt;ref&gt;Hess, R. (2007). The essential Blender: guide to 3D creation with the open source suite Blender. No Starch Press.&lt;/ref&gt;&lt;ref&gt;Mullen, T., Roosendaal, T., &amp; Kurdali, B. (2007). Introducing character animation with Blender. Wiley Pub.&lt;/ref&gt;
* '''[[/Hugin/|Hugin - Create an spherical image]]'''  for [https://aframe.io/examples/showcase/sky/ 360&lt;sup&gt;o&lt;/sup&gt;-Panorama Images]
* '''[[/AR.js/|AR.js - Augmented Reality in a browser with Marker Recognition]]''' with [https://www.youtube.com/watch?v=0MtvjFg7tik AR.js (Youtube Demo)]
* '''[[Photogrammetry/Regard3D|Regard3D - Create a 3D models from several images]]''' with [https://www.youtube.com/watch?v=GaYfpGcXxmA&amp;t=1499s Regard3D (Youtube Demo)] - (replace with WikiMedia Commons Videos if exists)
* '''[[/Threes.js/|Threes.js - Webbased 3D Modelling]]'''  - including animation and import of 3D models generated in [[/Blender/]]
* '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]''' Create a 360-degree image for Aframe (see [https://niebert.github.io/HuginSample 360 degree example])
* '''[[3D Modelling/Create 3D Models/AR.js|JSON3D4Aframe]]''' JSON Editor for AFrame and [[/AR.js/]]
* '''Create [[3D_Modelling/360_Degree_Video|360-degree Video]] with  [http://flimshaw.github.io/Valiant360/ Vailant360]''' and '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]'''.

== Learning Task ==
* '''([[Regard3D]])''' Create you own screencast of your 3D model you created with [[Regard3D]] (approx 10-15min as Walk Through) and provide the set of images on GitHub for other learners to work with (just as HuginSample for Hugin)
* '''([[/360-Degree AFrame/]])''' Create a 360 Degree image for AFrame with [https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]&lt;ref name=&quot;WikiCommons4AFrame&quot;&gt;WikiCommons4AFrame (2020) Bert Niehaus - [[AppLSAC]] for Wikiversity Learning Resource - Create 360-Degree Scenes in AFrame - URL: https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html (viewed/tested 2020/05/04)&lt;/ref&gt;
* '''([[/AR.js/]])''' Create a 3D model in JSON3D4Aframe and view the model on a Hiro marker. See also Mixare
* '''([[Markerless Tracking|Tracking]])''' Explore the concept of motion tracking with Javascript on https://www.trackingjs.com as OpenSource framework. Compare the approach with tracking approach on AR.js.

== External Resources ==
* '''[https://www.github.com/niebert/HuginSample HuginSample Github repository]''' with sample images of the [https://niebert.github.io/HuginSample Durlacher Saumarkt, Karlsruhe Durlach, Germany] 
* '''[http://hugin.sourceforge.net/ Hugin Download]:''' http://hugin.sourceforge.net/
* '''[https://www.gimp.org/downloads/ GIMP Download]''' for learning how to fix missing images with GIMP Clone Tool and projection fixing
* '''[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe]''' Webbased 3D Models constructed with geometric primitives (e.g. cubes, sphere, torus, rectangles, ...)
* '''[https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame2&quot;&gt;Bert Niehaus (2021) WikiCommons2Aframe - GitHub repository: https://github.com/niebert/WikiCommons2AFrame - Download: [https://github.com/niebert/WikiCommons2AFrame/archive/refs/heads/master.zip ZIP]&lt;/ref&gt; for using a 360 equirectangular image in Aframe as 360 degree images.

== Overview ==
{| class=&quot;wikitable&quot;
|-
! - !! JSON3DAFrame !! Hugin
|-
| Tool || 
[[File:JSON3D4Aframe_01.png|300px|thumb|[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
 || [[File:Hugin User Interface.png|300px|thumb|[http://hugin.sourceforge.net/Hugin Hugin Software] for 360 Degree Images in AFrame]]
|-
| Product || [[File:Water molecule  kanji.png|300px|thumb|Water Molecule with [[v:en:Augmented Reality|Augmented Reality]] AR.js on Kanji Marker - create with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe].]]
 || 
[[File:Aldara parks.jpg|300px|thumb|[https://niebert.github.com/HuginSample/Aldara_parks.html equirectangular Image from WikiMedia Commons rendered with Aframe as 360-Degree Image] (see [[v:en:3D Modelling/Create 3D Models/Hugin|Hugin]])]]

|}

== See also ==
* [[Markerless Tracking]]
* [[Digital Learning Environment]] 
* [[Open Innovation Ecosystem]]

== References ==

&lt;noinclude&gt; [[de:3D-Modellierung/3D-Modelle_erstellen]] &lt;/noinclude&gt;

[[Category:3D modeling]]</text>
      <sha1>bfiqjt0jhre8pnom197r6lnohvcodcd</sha1>
    </revision>
    <revision>
      <id>2805927</id>
      <parentid>2805926</parentid>
      <timestamp>2026-04-22T10:52:15Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* 3D Software Packages */</comment>
      <origin>2805927</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7903" sha1="pdefw6yalmj7mrpbbpigct4k80iud1o" xml:space="preserve">[[File:Water molecule aframe.png|300px|thumb|Water Molecule in Aframe - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Blender 2.79.png|thumb|350px|Blender Software - OpenSource - Windows, Linux, MacOSX]]
[[File:Water molecule kanji.png|300px|thumb|Water Molecule with [[Augmented Reality]] AR.js on Kanji Marker - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Sodium-chloride-3D-ionic.png|thumb|Crystal structure of sodium chloride (table salt) [https://niebert.github.io/JSON3D4Aframe/mods3d/cristal_lattice_aframe.html 3D Model in AFrame] created with [https://niebert.github.io/JSON3D4Aframe/ JSON3D4Aframe] ]]

The following learning resource supports you as an author to create you own three-dimensional models (see also demo 3D models provided by Lee Stemkoski&lt;ref&gt;Lee Stemkoski (2021) Aframe Examples on GitHub - Demo URL: https://stemkoski.github.io/A-Frame-Examples/ - GitHub repository: https://github.com/stemkoski/A-Frame-Examples [https://github.com/stemkoski/A-Frame-Examples/archive/refs/heads/master.zip ZIP] (accessed 2024/04/13) &lt;/ref&gt;. Provided tools are open source and can be used either for video animation three-dimensional space (e.g. genetic processes that are not visible in a microscope) are you construct 3-D models there are placed in the camera image of mobile device on specific markers (e.g. in [[3D_Modelling/Examples/|AR.js]]).

== Preparation ==
If you are following the [[Open Community Approach]] the created models can be adapted to local and regional requirements and constraints or specifically to target groups of learners that will use the models. When adaptation of models include specific information about the learner's environment then 3D models cannot be shared beyond the [[Digital Learning Environment]] in which the learner uses the open 3D model. In other use-cases the 3D model in the learning environment it is recommended to share the 3D models with the community of authors and learners on Wikiversity for further improvement (see also [[Open Innovation Ecosystem]]).
* currently 3D models cannot be integrated directly into Wikiversity, so files must be stored and shared on an other WebServer, that supports [[Version control|version control]] during the development process of the 3D models. 
* in this learning resource we use [https://www.github.com GitHub] and take advantage of the [[Version_control|version control system]] to share the files in the &lt;kbd&gt;/docs&lt;/kbd&gt; folder on GitHub. Feel free to use any other for the community available version control system. Version control is helpful to have the choice to revert changes due to drawbacks and challenges with the new version.
** Create a new [https://www.github.com GitHub] account if you do not have on (e.g. &quot;myacount&quot;).
** Login into [https://www.github.com GitHub] with your account.

== 3D Software Packages ==
* '''[[/360-Degree AFrame/]]''' Integrate 360 Images in Aframe for digital exploration of locations.
* '''[[/Wiki_360_Degree_AFrame/|360-Degree Image in Wikiversity]]''' Create a link to embed [[/360-Degree AFrame/|360-Degree Image]] from Wiki Commons in AFrame.
* '''[https://niebert.github.io/WikiCommons2AFrame/index.html WikiCommons2AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame&quot;/&gt;  Create 360-degree scenes in AFrame as a ZIP for offline (and online) use or create links with a 350-degree image that is online available as for online use. 
* '''[[/Blender/|Blender - Photorealistic 3D Modelling with Physics, Animation, ...]]:''' (e.g. Fluid Dynamics and Wind)&lt;ref&gt;Hess, R. (2007). The essential Blender: guide to 3D creation with the open source suite Blender. No Starch Press.&lt;/ref&gt;&lt;ref&gt;Mullen, T., Roosendaal, T., &amp; Kurdali, B. (2007). Introducing character animation with Blender. Wiley Pub.&lt;/ref&gt;
* '''[[/Hugin/|Hugin - Create an spherical image]]'''  for [https://aframe.io/examples/showcase/sky/ 360&lt;sup&gt;o&lt;/sup&gt;-Panorama Images]
* '''[[/AR.js/|AR.js - Augmented Reality in a browser with Marker Recognition]]''' with [https://www.youtube.com/watch?v=0MtvjFg7tik AR.js (Youtube Demo)]
* '''[[Photogrammetry/Regard3D|Regard3D - Create a 3D models from several images]]''' with [https://www.youtube.com/watch?v=GaYfpGcXxmA&amp;t=1499s Regard3D (Youtube Demo)] - (replace with WikiMedia Commons Videos if exists)
* '''[[/Threes.js/|Threes.js - Webbased 3D Modelling]]'''  - including animation and import of 3D models generated in [[/Blender/]]
* '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]''' Create a 360-degree image for Aframe (see [https://niebert.github.io/HuginSample 360 degree example])
* '''[[3D Modelling/Create 3D Models/AR.js|JSON3D4Aframe]]''' JSON Editor for AFrame and [[/AR.js/]]
* '''Create [[3D_Modelling/360_Degree_Video|360-degree Video]] with  [http://flimshaw.github.io/Valiant360/ Vailant360]''' and '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]'''.

== Learning Task ==
* '''([[Regard3D]])''' Create you own screencast of your 3D model you created with [[Regard3D]] (approx 10-15min as Walk Through) and provide the set of images on GitHub for other learners to work with (just as HuginSample for Hugin)
* '''([[/360-Degree AFrame/]])''' Create a 360 Degree image for AFrame with [https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]&lt;ref name=&quot;WikiCommons4AFrame&quot;&gt;WikiCommons4AFrame (2020) Bert Niehaus - [[AppLSAC]] for Wikiversity Learning Resource - Create 360-Degree Scenes in AFrame - URL: https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html (viewed/tested 2020/05/04)&lt;/ref&gt;
* '''([[/AR.js/]])''' Create a 3D model in JSON3D4Aframe and view the model on a Hiro marker. See also Mixare
* '''([[Markerless Tracking|Tracking]])''' Explore the concept of motion tracking with Javascript on https://www.trackingjs.com as OpenSource framework. Compare the approach with tracking approach on AR.js.

== External Resources ==
* '''[https://www.github.com/niebert/HuginSample HuginSample Github repository]''' with sample images of the [https://niebert.github.io/HuginSample Durlacher Saumarkt, Karlsruhe Durlach, Germany] 
* '''[http://hugin.sourceforge.net/ Hugin Download]:''' http://hugin.sourceforge.net/
* '''[https://www.gimp.org/downloads/ GIMP Download]''' for learning how to fix missing images with GIMP Clone Tool and projection fixing
* '''[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe]''' Webbased 3D Models constructed with geometric primitives (e.g. cubes, sphere, torus, rectangles, ...)
* '''[https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame2&quot;&gt;Bert Niehaus (2021) WikiCommons2Aframe - GitHub repository: https://github.com/niebert/WikiCommons2AFrame - Download: [https://github.com/niebert/WikiCommons2AFrame/archive/refs/heads/master.zip ZIP]&lt;/ref&gt; for using a 360 equirectangular image in Aframe as 360 degree images.

== Overview ==
{| class=&quot;wikitable&quot;
|-
! - !! JSON3DAFrame !! Hugin
|-
| Tool || 
[[File:JSON3D4Aframe_01.png|300px|thumb|[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
 || [[File:Hugin User Interface.png|300px|thumb|[http://hugin.sourceforge.net/Hugin Hugin Software] for 360 Degree Images in AFrame]]
|-
| Product || [[File:Water molecule  kanji.png|300px|thumb|Water Molecule with [[v:en:Augmented Reality|Augmented Reality]] AR.js on Kanji Marker - create with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe].]]
 || 
[[File:Aldara parks.jpg|300px|thumb|[https://niebert.github.com/HuginSample/Aldara_parks.html equirectangular Image from WikiMedia Commons rendered with Aframe as 360-Degree Image] (see [[v:en:3D Modelling/Create 3D Models/Hugin|Hugin]])]]

|}

== See also ==
* [[Markerless Tracking]]
* [[Digital Learning Environment]] 
* [[Open Innovation Ecosystem]]

== References ==

&lt;noinclude&gt; [[de:3D-Modellierung/3D-Modelle_erstellen]] &lt;/noinclude&gt;

[[Category:3D modeling]]</text>
      <sha1>pdefw6yalmj7mrpbbpigct4k80iud1o</sha1>
    </revision>
    <revision>
      <id>2805928</id>
      <parentid>2805927</parentid>
      <timestamp>2026-04-22T10:55:28Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* 3D Software Packages */</comment>
      <origin>2805928</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7958" sha1="rl74bk60lqqbrtyz8txaf64dku64by3" xml:space="preserve">[[File:Water molecule aframe.png|300px|thumb|Water Molecule in Aframe - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Blender 2.79.png|thumb|350px|Blender Software - OpenSource - Windows, Linux, MacOSX]]
[[File:Water molecule kanji.png|300px|thumb|Water Molecule with [[Augmented Reality]] AR.js on Kanji Marker - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Sodium-chloride-3D-ionic.png|thumb|Crystal structure of sodium chloride (table salt) [https://niebert.github.io/JSON3D4Aframe/mods3d/cristal_lattice_aframe.html 3D Model in AFrame] created with [https://niebert.github.io/JSON3D4Aframe/ JSON3D4Aframe] ]]

The following learning resource supports you as an author to create you own three-dimensional models (see also demo 3D models provided by Lee Stemkoski&lt;ref&gt;Lee Stemkoski (2021) Aframe Examples on GitHub - Demo URL: https://stemkoski.github.io/A-Frame-Examples/ - GitHub repository: https://github.com/stemkoski/A-Frame-Examples [https://github.com/stemkoski/A-Frame-Examples/archive/refs/heads/master.zip ZIP] (accessed 2024/04/13) &lt;/ref&gt;. Provided tools are open source and can be used either for video animation three-dimensional space (e.g. genetic processes that are not visible in a microscope) are you construct 3-D models there are placed in the camera image of mobile device on specific markers (e.g. in [[3D_Modelling/Examples/|AR.js]]).

== Preparation ==
If you are following the [[Open Community Approach]] the created models can be adapted to local and regional requirements and constraints or specifically to target groups of learners that will use the models. When adaptation of models include specific information about the learner's environment then 3D models cannot be shared beyond the [[Digital Learning Environment]] in which the learner uses the open 3D model. In other use-cases the 3D model in the learning environment it is recommended to share the 3D models with the community of authors and learners on Wikiversity for further improvement (see also [[Open Innovation Ecosystem]]).
* currently 3D models cannot be integrated directly into Wikiversity, so files must be stored and shared on an other WebServer, that supports [[Version control|version control]] during the development process of the 3D models. 
* in this learning resource we use [https://www.github.com GitHub] and take advantage of the [[Version_control|version control system]] to share the files in the &lt;kbd&gt;/docs&lt;/kbd&gt; folder on GitHub. Feel free to use any other for the community available version control system. Version control is helpful to have the choice to revert changes due to drawbacks and challenges with the new version.
** Create a new [https://www.github.com GitHub] account if you do not have on (e.g. &quot;myacount&quot;).
** Login into [https://www.github.com GitHub] with your account.

== 3D Software Packages ==
* '''[[/360-Degree AFrame/]]''' Integrate 360 Images in Aframe for digital exploration of locations.
* '''[[/Wiki_360_Degree_AFrame/|360-Degree Image in Wikiversity]]''' Create a link to embed [[/360-Degree AFrame/|360-Degree Image]] from Wiki Commons in AFrame.
* '''[https://niebert.github.io/WikiCommons2AFrame/index.html WikiCommons2AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame&quot;/&gt;  Create 360-degree scenes in AFrame as a ZIP for offline (and online) use or create links with a list of 360-degree image used as support material for exploration of a location e.g. for (e.g. [[Real World Lab]]). 
* '''[[/Blender/|Blender - Photorealistic 3D Modelling with Physics, Animation, ...]]:''' (e.g. Fluid Dynamics and Wind)&lt;ref&gt;Hess, R. (2007). The essential Blender: guide to 3D creation with the open source suite Blender. No Starch Press.&lt;/ref&gt;&lt;ref&gt;Mullen, T., Roosendaal, T., &amp; Kurdali, B. (2007). Introducing character animation with Blender. Wiley Pub.&lt;/ref&gt;
* '''[[/Hugin/|Hugin - Create an spherical image]]'''  for [https://aframe.io/examples/showcase/sky/ 360&lt;sup&gt;o&lt;/sup&gt;-Panorama Images]
* '''[[/AR.js/|AR.js - Augmented Reality in a browser with Marker Recognition]]''' with [https://www.youtube.com/watch?v=0MtvjFg7tik AR.js (Youtube Demo)]
* '''[[Photogrammetry/Regard3D|Regard3D - Create a 3D models from several images]]''' with [https://www.youtube.com/watch?v=GaYfpGcXxmA&amp;t=1499s Regard3D (Youtube Demo)] - (replace with WikiMedia Commons Videos if exists)
* '''[[/Threes.js/|Threes.js - Webbased 3D Modelling]]'''  - including animation and import of 3D models generated in [[/Blender/]]
* '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]''' Create a 360-degree image for Aframe (see [https://niebert.github.io/HuginSample 360 degree example])
* '''[[3D Modelling/Create 3D Models/AR.js|JSON3D4Aframe]]''' JSON Editor for AFrame and [[/AR.js/]]
* '''Create [[3D_Modelling/360_Degree_Video|360-degree Video]] with  [http://flimshaw.github.io/Valiant360/ Vailant360]''' and '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]'''.

== Learning Task ==
* '''([[Regard3D]])''' Create you own screencast of your 3D model you created with [[Regard3D]] (approx 10-15min as Walk Through) and provide the set of images on GitHub for other learners to work with (just as HuginSample for Hugin)
* '''([[/360-Degree AFrame/]])''' Create a 360 Degree image for AFrame with [https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]&lt;ref name=&quot;WikiCommons4AFrame&quot;&gt;WikiCommons4AFrame (2020) Bert Niehaus - [[AppLSAC]] for Wikiversity Learning Resource - Create 360-Degree Scenes in AFrame - URL: https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html (viewed/tested 2020/05/04)&lt;/ref&gt;
* '''([[/AR.js/]])''' Create a 3D model in JSON3D4Aframe and view the model on a Hiro marker. See also Mixare
* '''([[Markerless Tracking|Tracking]])''' Explore the concept of motion tracking with Javascript on https://www.trackingjs.com as OpenSource framework. Compare the approach with tracking approach on AR.js.

== External Resources ==
* '''[https://www.github.com/niebert/HuginSample HuginSample Github repository]''' with sample images of the [https://niebert.github.io/HuginSample Durlacher Saumarkt, Karlsruhe Durlach, Germany] 
* '''[http://hugin.sourceforge.net/ Hugin Download]:''' http://hugin.sourceforge.net/
* '''[https://www.gimp.org/downloads/ GIMP Download]''' for learning how to fix missing images with GIMP Clone Tool and projection fixing
* '''[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe]''' Webbased 3D Models constructed with geometric primitives (e.g. cubes, sphere, torus, rectangles, ...)
* '''[https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame2&quot;&gt;Bert Niehaus (2021) WikiCommons2Aframe - GitHub repository: https://github.com/niebert/WikiCommons2AFrame - Download: [https://github.com/niebert/WikiCommons2AFrame/archive/refs/heads/master.zip ZIP]&lt;/ref&gt; for using a 360 equirectangular image in Aframe as 360 degree images.

== Overview ==
{| class=&quot;wikitable&quot;
|-
! - !! JSON3DAFrame !! Hugin
|-
| Tool || 
[[File:JSON3D4Aframe_01.png|300px|thumb|[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
 || [[File:Hugin User Interface.png|300px|thumb|[http://hugin.sourceforge.net/Hugin Hugin Software] for 360 Degree Images in AFrame]]
|-
| Product || [[File:Water molecule  kanji.png|300px|thumb|Water Molecule with [[v:en:Augmented Reality|Augmented Reality]] AR.js on Kanji Marker - create with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe].]]
 || 
[[File:Aldara parks.jpg|300px|thumb|[https://niebert.github.com/HuginSample/Aldara_parks.html equirectangular Image from WikiMedia Commons rendered with Aframe as 360-Degree Image] (see [[v:en:3D Modelling/Create 3D Models/Hugin|Hugin]])]]

|}

== See also ==
* [[Markerless Tracking]]
* [[Digital Learning Environment]] 
* [[Open Innovation Ecosystem]]

== References ==

&lt;noinclude&gt; [[de:3D-Modellierung/3D-Modelle_erstellen]] &lt;/noinclude&gt;

[[Category:3D modeling]]</text>
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    </revision>
    <revision>
      <id>2805929</id>
      <parentid>2805928</parentid>
      <timestamp>2026-04-22T10:56:17Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* See also */</comment>
      <origin>2805929</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7980" sha1="pjm09y81yrt9218otwdsxypg1jj6ei9" xml:space="preserve">[[File:Water molecule aframe.png|300px|thumb|Water Molecule in Aframe - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Blender 2.79.png|thumb|350px|Blender Software - OpenSource - Windows, Linux, MacOSX]]
[[File:Water molecule kanji.png|300px|thumb|Water Molecule with [[Augmented Reality]] AR.js on Kanji Marker - created with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
[[File:Sodium-chloride-3D-ionic.png|thumb|Crystal structure of sodium chloride (table salt) [https://niebert.github.io/JSON3D4Aframe/mods3d/cristal_lattice_aframe.html 3D Model in AFrame] created with [https://niebert.github.io/JSON3D4Aframe/ JSON3D4Aframe] ]]

The following learning resource supports you as an author to create you own three-dimensional models (see also demo 3D models provided by Lee Stemkoski&lt;ref&gt;Lee Stemkoski (2021) Aframe Examples on GitHub - Demo URL: https://stemkoski.github.io/A-Frame-Examples/ - GitHub repository: https://github.com/stemkoski/A-Frame-Examples [https://github.com/stemkoski/A-Frame-Examples/archive/refs/heads/master.zip ZIP] (accessed 2024/04/13) &lt;/ref&gt;. Provided tools are open source and can be used either for video animation three-dimensional space (e.g. genetic processes that are not visible in a microscope) are you construct 3-D models there are placed in the camera image of mobile device on specific markers (e.g. in [[3D_Modelling/Examples/|AR.js]]).

== Preparation ==
If you are following the [[Open Community Approach]] the created models can be adapted to local and regional requirements and constraints or specifically to target groups of learners that will use the models. When adaptation of models include specific information about the learner's environment then 3D models cannot be shared beyond the [[Digital Learning Environment]] in which the learner uses the open 3D model. In other use-cases the 3D model in the learning environment it is recommended to share the 3D models with the community of authors and learners on Wikiversity for further improvement (see also [[Open Innovation Ecosystem]]).
* currently 3D models cannot be integrated directly into Wikiversity, so files must be stored and shared on an other WebServer, that supports [[Version control|version control]] during the development process of the 3D models. 
* in this learning resource we use [https://www.github.com GitHub] and take advantage of the [[Version_control|version control system]] to share the files in the &lt;kbd&gt;/docs&lt;/kbd&gt; folder on GitHub. Feel free to use any other for the community available version control system. Version control is helpful to have the choice to revert changes due to drawbacks and challenges with the new version.
** Create a new [https://www.github.com GitHub] account if you do not have on (e.g. &quot;myacount&quot;).
** Login into [https://www.github.com GitHub] with your account.

== 3D Software Packages ==
* '''[[/360-Degree AFrame/]]''' Integrate 360 Images in Aframe for digital exploration of locations.
* '''[[/Wiki_360_Degree_AFrame/|360-Degree Image in Wikiversity]]''' Create a link to embed [[/360-Degree AFrame/|360-Degree Image]] from Wiki Commons in AFrame.
* '''[https://niebert.github.io/WikiCommons2AFrame/index.html WikiCommons2AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame&quot;/&gt;  Create 360-degree scenes in AFrame as a ZIP for offline (and online) use or create links with a list of 360-degree image used as support material for exploration of a location e.g. for (e.g. [[Real World Lab]]). 
* '''[[/Blender/|Blender - Photorealistic 3D Modelling with Physics, Animation, ...]]:''' (e.g. Fluid Dynamics and Wind)&lt;ref&gt;Hess, R. (2007). The essential Blender: guide to 3D creation with the open source suite Blender. No Starch Press.&lt;/ref&gt;&lt;ref&gt;Mullen, T., Roosendaal, T., &amp; Kurdali, B. (2007). Introducing character animation with Blender. Wiley Pub.&lt;/ref&gt;
* '''[[/Hugin/|Hugin - Create an spherical image]]'''  for [https://aframe.io/examples/showcase/sky/ 360&lt;sup&gt;o&lt;/sup&gt;-Panorama Images]
* '''[[/AR.js/|AR.js - Augmented Reality in a browser with Marker Recognition]]''' with [https://www.youtube.com/watch?v=0MtvjFg7tik AR.js (Youtube Demo)]
* '''[[Photogrammetry/Regard3D|Regard3D - Create a 3D models from several images]]''' with [https://www.youtube.com/watch?v=GaYfpGcXxmA&amp;t=1499s Regard3D (Youtube Demo)] - (replace with WikiMedia Commons Videos if exists)
* '''[[/Threes.js/|Threes.js - Webbased 3D Modelling]]'''  - including animation and import of 3D models generated in [[/Blender/]]
* '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]''' Create a 360-degree image for Aframe (see [https://niebert.github.io/HuginSample 360 degree example])
* '''[[3D Modelling/Create 3D Models/AR.js|JSON3D4Aframe]]''' JSON Editor for AFrame and [[/AR.js/]]
* '''Create [[3D_Modelling/360_Degree_Video|360-degree Video]] with  [http://flimshaw.github.io/Valiant360/ Vailant360]''' and '''[[3D Modelling/Create 3D Models/Hugin|Hugin]]'''.

== Learning Task ==
* '''([[Regard3D]])''' Create you own screencast of your 3D model you created with [[Regard3D]] (approx 10-15min as Walk Through) and provide the set of images on GitHub for other learners to work with (just as HuginSample for Hugin)
* '''([[/360-Degree AFrame/]])''' Create a 360 Degree image for AFrame with [https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]&lt;ref name=&quot;WikiCommons4AFrame&quot;&gt;WikiCommons4AFrame (2020) Bert Niehaus - [[AppLSAC]] for Wikiversity Learning Resource - Create 360-Degree Scenes in AFrame - URL: https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html (viewed/tested 2020/05/04)&lt;/ref&gt;
* '''([[/AR.js/]])''' Create a 3D model in JSON3D4Aframe and view the model on a Hiro marker. See also Mixare
* '''([[Markerless Tracking|Tracking]])''' Explore the concept of motion tracking with Javascript on https://www.trackingjs.com as OpenSource framework. Compare the approach with tracking approach on AR.js.

== External Resources ==
* '''[https://www.github.com/niebert/HuginSample HuginSample Github repository]''' with sample images of the [https://niebert.github.io/HuginSample Durlacher Saumarkt, Karlsruhe Durlach, Germany] 
* '''[http://hugin.sourceforge.net/ Hugin Download]:''' http://hugin.sourceforge.net/
* '''[https://www.gimp.org/downloads/ GIMP Download]''' for learning how to fix missing images with GIMP Clone Tool and projection fixing
* '''[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe]''' Webbased 3D Models constructed with geometric primitives (e.g. cubes, sphere, torus, rectangles, ...)
* '''[https://niebert.github.io/WikiCommons2AFrame/aframe360starter.html WikiCommons4AFrame]'''&lt;ref name=&quot;WikiCommons4AFrame2&quot;&gt;Bert Niehaus (2021) WikiCommons2Aframe - GitHub repository: https://github.com/niebert/WikiCommons2AFrame - Download: [https://github.com/niebert/WikiCommons2AFrame/archive/refs/heads/master.zip ZIP]&lt;/ref&gt; for using a 360 equirectangular image in Aframe as 360 degree images.

== Overview ==
{| class=&quot;wikitable&quot;
|-
! - !! JSON3DAFrame !! Hugin
|-
| Tool || 
[[File:JSON3D4Aframe_01.png|300px|thumb|[https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] ]]
 || [[File:Hugin User Interface.png|300px|thumb|[http://hugin.sourceforge.net/Hugin Hugin Software] for 360 Degree Images in AFrame]]
|-
| Product || [[File:Water molecule  kanji.png|300px|thumb|Water Molecule with [[v:en:Augmented Reality|Augmented Reality]] AR.js on Kanji Marker - create with [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe].]]
 || 
[[File:Aldara parks.jpg|300px|thumb|[https://niebert.github.com/HuginSample/Aldara_parks.html equirectangular Image from WikiMedia Commons rendered with Aframe as 360-Degree Image] (see [[v:en:3D Modelling/Create 3D Models/Hugin|Hugin]])]]

|}


== See also ==
* [[Markerless Tracking]]
* [[Digital Learning Environment]] 
* [[Open Innovation Ecosystem]]
* [[Real World Lab]]

== References ==

&lt;noinclude&gt; [[de:3D-Modellierung/3D-Modelle_erstellen]] &lt;/noinclude&gt;

[[Category:3D modeling]]</text>
      <sha1>pjm09y81yrt9218otwdsxypg1jj6ei9</sha1>
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  </page>
  <page>
    <title>3D Modelling/Examples/Panorama 360</title>
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    <revision>
      <id>2805932</id>
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      <timestamp>2026-04-22T11:37:54Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* See also */</comment>
      <origin>2805932</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8801" sha1="p6f5uykhm8wpkdkonbpgow2o1v0h5yo" xml:space="preserve">[[File:aframe360image.png|thumb|300px|[https://aframe.io/ AFrame 360 Image]  - Look around by draging mouse with mouse button pressed]]
[[File:Eusserthal aframe panorama.png|thumb|300px|[https://aframe.io/examples/showcase/sky/ AFrame 360 Image]  - Look around by draging mouse with mouse button pressed]]


[[File:Hugin result in aframe.png|thumb|300px|[https://niebert.github.io/HuginSample/ AFrame Example Durlach] or [https://aframe.io/examples/showcase/sky/ AFrame Sample 360 Degree Image]  - Look around by draging mouse with mouse button pressed]]
For the Panoramas with 360 Degree view there are to different geometric option to create 360-Degree view for a geographical location:
* '''Tube View:''' Look around parallel to the ground an just turn 360 degree. Smartphone can be used to create 360 degree image.
* '''Sphere View:''' In addition to 360 degree turn, you can [https://niebert.github.io/HuginSample/ look up and down] with an [[equirectangular projection]]. To create spherical project, you need e.g. the [http://hugin.sourceforge.net/download/ OpenSource software Hugin to create image], that can be used for a spherical projection. The generated image by Hugin viewed in a [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg browser seems to have a distortion] which provides in AFrame the desired [[w:en:Projective_geometry|projective geometry]].

== Learning Task==
* '''(VR-Headset)''' If you have access to a VR-Headset display the [https://aframe.io/examples/showcase/sky/ AFrame demo with a 360 degree image] and press the VR-Headset Icon at the bottom right.
* '''(Projection)''' Explain why there is the distortion in the image projection, when you do not view the images outside from the [https://aframe.io/examples/showcase/sky/ Aframe Environment] (see [https://aframe.io/aframe/examples/boilerplate/panorama/puydesancy.jpg Source Image of Puy de Sancy]).
* '''(Stiching 360-Panorama from single Images)''' Create your [[3D_Modelling/Create_3D_Models/Hugin|own Panorama with OpenSource image stiching software Hugin]] and view the panorama in AFrame.
* '''([[Real World Lab]])''' Explore the concept of [[Real World Lab|Real World Labs]] and explain the concept of 360 degree panorama images for preparing a visit in the [[Real World Lab]]!
* '''(Panoviewer for WikiCommons Images)''' You can use [https://panoviewer.toolforge.org/ PanoViewer] for visualisation of 360-Degree Image in Wikiversity and Wikipedia.
** '''Wikipedia Article:''' [[w:en:Jelgava_Palace|Jelgava Palace]]
** '''WikiCommons Source Image:''' 
*** [[c:File:Jelgavas_pils.jpg|Jelgavas_pils.jpg]]
*** [[c:File:Jelgavas_pils.jpg|Durlach_saumarkt.jpg]]
** '''360-Degree Image in Panoviewer'''
*** '''360-Degree Image ''[https://panoviewer.toolforge.org/#Jelgavas_pils.jpg Jegalvas Palace]'':''' https://panoviewer.toolforge.org/#Jelgavas_pils.jpg Jegalvas Palace
*** '''360-Degree Image ''[https://panoviewer.toolforge.org/#Durlach_saumarkt.jpg Durlach Saumarkt]'':''' https://panoviewer.toolforge.org/#Durlach_saumarkt.jpg Durlach Saumarkt

== List of Examples ==
See also [[3D Modelling]] and [https://niebert.github.io/WikiCommons2AFrame/ WikiCommons2AFrame] for creating your own 360 degree panoramas with equirectangular images.

=== Nature: River Rhine near Cologne/Germany ===
The following 360 Degree image are displayed with Aframe which make them usable with VR-headsets.
* [https://aframe.io/examples/showcase/sky/ Aframe Demo: Puy de Sancy - France]
* [https://niebert.github.io/HuginSample/rhein1_rodenkirchen.html River Rhine 1: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein2_rodenkirchen.html River Rhine 2: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein3_rodenkirchen.html River Rhine 3: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein4_rodenkirchen.html River Rhine 4: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/Aldara_parks.html Aldara Parks / ] 

=== Cities ===
* [https://niebert.github.io/HuginSample/durlach_saumarkt.html Durlach Saumarkt / Karlsruhe / Germany] - (see also [[Photogrammetry/Regard3D#Karlsburg_Images_and_Models|Schloss Durlach Photogrammetry]])

=== Trees 3D ===
* [http://www.snappytree.com 3D-Models for Trees]&lt;ref&gt; Paul Brunt (2012) Open-Source-Software  SnappyTree - GitHub-Repository: https://github.com/supereggbert/SnappyTree/ Demo-3D-Trees: http://www.snappytree.com (Zugriff 2021/05/07).&lt;/ref&gt; - [https://github.com/supereggbert/SnappyTree/ GitHub-Repository Treegenerator Snappy-Tree - OpenSource] for integration in 360-degree panoramas.

== Tube View ==
[[File:Blender-nurbs-tube.png|thumb|Tube Projection]]
The following examples start with the tube view and proceed to more advanced projections with Panorama Image Stitching for sphere view of an panorama.
* '''(Tube Panorama View)''' Explore a [https://niebert.github.io/panorama360/view360.html 360 degree webbased panorama] projected on plane (move right or left).
* '''(Image Selection)''' Search for [https://commons.wikimedia.org/wiki/Category:Panoramas panorama images on WikiMedia] and create an  panorama 360 image with the filename of the image with the [https://niebert.github.io/panorama360 Panorama 360 Creator]!
* '''([https://niebert.github.io/panorama360/regensburg360.html Clickable Areas in Panorama])''' If you want to create an [[Image Map]] for the Panorama use the [https://niebert.github.io/imgmap Online Image Map creator] and switch output format to &lt;kbd&gt;Panorama 360 imagemap&lt;/kbd&gt; in select box.

== Sphere View Panorama ==
[[File:Equirectangular sphere.png|thumb|Suface of sphere projected to a rectangle - equirectangular image]]
[[File:Equirectangular-projection.jpg|thumb|Equirectangular Projection of the surface of the earth]]
The first step is to create a multiple view of a selected location and take pictures from a centre of view into all direction with overlapping images (1/4 overlap). The [http://hugin.sourceforge.net/ Panorama Image Stitching Software Hugin] support you in creating the 360-degree spherical view for the panorama. 
* '''(Final Product of Learning Task)''' Explore the [https://niebert.github.io/HuginSample/index.html final product of the 360&lt;sup&gt;o&lt;/sup&gt; Example] first.
* '''(Learning Task)''' Learning task is to create the [https://niebert.github.io/HuginSample/index.html final product] from the collection of images in the [https://github.com/niebert/HuginSample HuginSample GitHub repository] and learn how to fix some missing images for the spherical view.
* '''(Demo Image Set as ZIP)''' The [https://github.com/niebert/HuginSample/archive/master.zip ZIP-file] of all images and the final product can be downloaded form [https://github.com/niebert/HuginSample Hugin Sample Repository on Github] ([https://github.com/niebert/HuginSample/archive/master.zip Direct Download Image ZIP]).
* '''(Perform Hugin Photo Stitching for Panorama)''' The [https://niebert.github.io/HuginSample/index.html final product uses the 360-Degree image of AFrame]. To learn about semiautomated creation of spherical projections of multiple camera images, proceed with the information on [https://github.com/niebert/HuginSample HuginSample README.md]. 
* '''(Hugin Tutorial Video)''' [https://www.youtube.com/watch?v=O_gONzUndQo Youtube Video by Brian Cluff about Hugin] - 2017/02/07
* '''(AFrame Background)''' More examples and information about AFrame can be found on [https://aframe.io/examples/showcase/sky/ AFrame.io Website].

=== Spherical (equirectangular) Images ===
You can explore existing spherical (equirectangular) images in [https://commons.wikimedia.org/wiki/Category:Equirectangular_projection WikiMedia Commons].

== See also ==
* [[3D_Modelling/Create_3D_Models/Hugin|Open Source Software Hugin]]
* [https://aframe.io/examples AFrame]
* [https://niebert.github.io/WikiCommons2AFrame WikiCommons4AFrame] - click on AFrame settings and select the spherical panorama as background for the 3D Models.
* [https://commons.wikimedia.org/wiki/Category:Spherical_panoramics Spherical Panorama Images] - available von WikiMedia Commons, that can be used as 360 Degree Images in AFrame with a-sky tag.
* [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] - click on AFrame settings and select the spherical panorama as background for the 3D Models.
* [[Real World Lab]] with 360-Degree-Images.
* [[3D_Modelling/Create_3D_Models/360-Degree_AFrame|Create AFrame Scenes with 360-Degree Images]]
* [[Image_Map/Tutorial|Image Map Tutorial for learning environments in Wikiversity]]
* [[Photogrammetry]]
* [[Equirectangular projection]]

== References ==

[[Category:3D modeling]]
[[Category:AFrame]]

&lt;noinclude&gt;[[de:3D-Modellierung/Beispiele/360-Grad-Panorama]]&lt;/noinclude&gt;</text>
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      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
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      <comment>/* See also */</comment>
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      <text bytes="8801" sha1="h9303i4vb7edasdx3kon7edw8l2u0qa" xml:space="preserve">[[File:aframe360image.png|thumb|300px|[https://aframe.io/ AFrame 360 Image]  - Look around by draging mouse with mouse button pressed]]
[[File:Eusserthal aframe panorama.png|thumb|300px|[https://aframe.io/examples/showcase/sky/ AFrame 360 Image]  - Look around by draging mouse with mouse button pressed]]


[[File:Hugin result in aframe.png|thumb|300px|[https://niebert.github.io/HuginSample/ AFrame Example Durlach] or [https://aframe.io/examples/showcase/sky/ AFrame Sample 360 Degree Image]  - Look around by draging mouse with mouse button pressed]]
For the Panoramas with 360 Degree view there are to different geometric option to create 360-Degree view for a geographical location:
* '''Tube View:''' Look around parallel to the ground an just turn 360 degree. Smartphone can be used to create 360 degree image.
* '''Sphere View:''' In addition to 360 degree turn, you can [https://niebert.github.io/HuginSample/ look up and down] with an [[equirectangular projection]]. To create spherical project, you need e.g. the [http://hugin.sourceforge.net/download/ OpenSource software Hugin to create image], that can be used for a spherical projection. The generated image by Hugin viewed in a [https://niebert.github.io/HuginSample/img/durlach_saumarkt.jpg browser seems to have a distortion] which provides in AFrame the desired [[w:en:Projective_geometry|projective geometry]].

== Learning Task==
* '''(VR-Headset)''' If you have access to a VR-Headset display the [https://aframe.io/examples/showcase/sky/ AFrame demo with a 360 degree image] and press the VR-Headset Icon at the bottom right.
* '''(Projection)''' Explain why there is the distortion in the image projection, when you do not view the images outside from the [https://aframe.io/examples/showcase/sky/ Aframe Environment] (see [https://aframe.io/aframe/examples/boilerplate/panorama/puydesancy.jpg Source Image of Puy de Sancy]).
* '''(Stiching 360-Panorama from single Images)''' Create your [[3D_Modelling/Create_3D_Models/Hugin|own Panorama with OpenSource image stiching software Hugin]] and view the panorama in AFrame.
* '''([[Real World Lab]])''' Explore the concept of [[Real World Lab|Real World Labs]] and explain the concept of 360 degree panorama images for preparing a visit in the [[Real World Lab]]!
* '''(Panoviewer for WikiCommons Images)''' You can use [https://panoviewer.toolforge.org/ PanoViewer] for visualisation of 360-Degree Image in Wikiversity and Wikipedia.
** '''Wikipedia Article:''' [[w:en:Jelgava_Palace|Jelgava Palace]]
** '''WikiCommons Source Image:''' 
*** [[c:File:Jelgavas_pils.jpg|Jelgavas_pils.jpg]]
*** [[c:File:Jelgavas_pils.jpg|Durlach_saumarkt.jpg]]
** '''360-Degree Image in Panoviewer'''
*** '''360-Degree Image ''[https://panoviewer.toolforge.org/#Jelgavas_pils.jpg Jegalvas Palace]'':''' https://panoviewer.toolforge.org/#Jelgavas_pils.jpg Jegalvas Palace
*** '''360-Degree Image ''[https://panoviewer.toolforge.org/#Durlach_saumarkt.jpg Durlach Saumarkt]'':''' https://panoviewer.toolforge.org/#Durlach_saumarkt.jpg Durlach Saumarkt

== List of Examples ==
See also [[3D Modelling]] and [https://niebert.github.io/WikiCommons2AFrame/ WikiCommons2AFrame] for creating your own 360 degree panoramas with equirectangular images.

=== Nature: River Rhine near Cologne/Germany ===
The following 360 Degree image are displayed with Aframe which make them usable with VR-headsets.
* [https://aframe.io/examples/showcase/sky/ Aframe Demo: Puy de Sancy - France]
* [https://niebert.github.io/HuginSample/rhein1_rodenkirchen.html River Rhine 1: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein2_rodenkirchen.html River Rhine 2: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein3_rodenkirchen.html River Rhine 3: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/rhein4_rodenkirchen.html River Rhine 4: near Cologne / Rodenkirchen / Germany] 
* [https://niebert.github.io/HuginSample/Aldara_parks.html Aldara Parks / ] 

=== Cities ===
* [https://niebert.github.io/HuginSample/durlach_saumarkt.html Durlach Saumarkt / Karlsruhe / Germany] - (see also [[Photogrammetry/Regard3D#Karlsburg_Images_and_Models|Schloss Durlach Photogrammetry]])

=== Trees 3D ===
* [http://www.snappytree.com 3D-Models for Trees]&lt;ref&gt; Paul Brunt (2012) Open-Source-Software  SnappyTree - GitHub-Repository: https://github.com/supereggbert/SnappyTree/ Demo-3D-Trees: http://www.snappytree.com (Zugriff 2021/05/07).&lt;/ref&gt; - [https://github.com/supereggbert/SnappyTree/ GitHub-Repository Treegenerator Snappy-Tree - OpenSource] for integration in 360-degree panoramas.

== Tube View ==
[[File:Blender-nurbs-tube.png|thumb|Tube Projection]]
The following examples start with the tube view and proceed to more advanced projections with Panorama Image Stitching for sphere view of an panorama.
* '''(Tube Panorama View)''' Explore a [https://niebert.github.io/panorama360/view360.html 360 degree webbased panorama] projected on plane (move right or left).
* '''(Image Selection)''' Search for [https://commons.wikimedia.org/wiki/Category:Panoramas panorama images on WikiMedia] and create an  panorama 360 image with the filename of the image with the [https://niebert.github.io/panorama360 Panorama 360 Creator]!
* '''([https://niebert.github.io/panorama360/regensburg360.html Clickable Areas in Panorama])''' If you want to create an [[Image Map]] for the Panorama use the [https://niebert.github.io/imgmap Online Image Map creator] and switch output format to &lt;kbd&gt;Panorama 360 imagemap&lt;/kbd&gt; in select box.

== Sphere View Panorama ==
[[File:Equirectangular sphere.png|thumb|Suface of sphere projected to a rectangle - equirectangular image]]
[[File:Equirectangular-projection.jpg|thumb|Equirectangular Projection of the surface of the earth]]
The first step is to create a multiple view of a selected location and take pictures from a centre of view into all direction with overlapping images (1/4 overlap). The [http://hugin.sourceforge.net/ Panorama Image Stitching Software Hugin] support you in creating the 360-degree spherical view for the panorama. 
* '''(Final Product of Learning Task)''' Explore the [https://niebert.github.io/HuginSample/index.html final product of the 360&lt;sup&gt;o&lt;/sup&gt; Example] first.
* '''(Learning Task)''' Learning task is to create the [https://niebert.github.io/HuginSample/index.html final product] from the collection of images in the [https://github.com/niebert/HuginSample HuginSample GitHub repository] and learn how to fix some missing images for the spherical view.
* '''(Demo Image Set as ZIP)''' The [https://github.com/niebert/HuginSample/archive/master.zip ZIP-file] of all images and the final product can be downloaded form [https://github.com/niebert/HuginSample Hugin Sample Repository on Github] ([https://github.com/niebert/HuginSample/archive/master.zip Direct Download Image ZIP]).
* '''(Perform Hugin Photo Stitching for Panorama)''' The [https://niebert.github.io/HuginSample/index.html final product uses the 360-Degree image of AFrame]. To learn about semiautomated creation of spherical projections of multiple camera images, proceed with the information on [https://github.com/niebert/HuginSample HuginSample README.md]. 
* '''(Hugin Tutorial Video)''' [https://www.youtube.com/watch?v=O_gONzUndQo Youtube Video by Brian Cluff about Hugin] - 2017/02/07
* '''(AFrame Background)''' More examples and information about AFrame can be found on [https://aframe.io/examples/showcase/sky/ AFrame.io Website].

=== Spherical (equirectangular) Images ===
You can explore existing spherical (equirectangular) images in [https://commons.wikimedia.org/wiki/Category:Equirectangular_projection WikiMedia Commons].

== See also ==
* [[3D_Modelling/Create_3D_Models/Hugin|Open Source Software Hugin]]
* [https://aframe.io/examples AFrame]
* [https://niebert.github.io/WikiCommons2AFrame WikiCommons2AFrame] - click on AFrame settings and select the spherical panorama as background for the 3D Models.
* [https://commons.wikimedia.org/wiki/Category:Spherical_panoramics Spherical Panorama Images] - available von WikiMedia Commons, that can be used as 360 Degree Images in AFrame with a-sky tag.
* [https://niebert.github.io/JSON3D4Aframe JSON3D4Aframe] - click on AFrame settings and select the spherical panorama as background for the 3D Models.
* [[Real World Lab]] with 360-Degree-Images.
* [[3D_Modelling/Create_3D_Models/360-Degree_AFrame|Create AFrame Scenes with 360-Degree Images]]
* [[Image_Map/Tutorial|Image Map Tutorial for learning environments in Wikiversity]]
* [[Photogrammetry]]
* [[Equirectangular projection]]

== References ==

[[Category:3D modeling]]
[[Category:AFrame]]

&lt;noinclude&gt;[[de:3D-Modellierung/Beispiele/360-Grad-Panorama]]&lt;/noinclude&gt;</text>
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    <title>Occupational Epidemiology/Research tools/Theoretical Program - open here/Electronic Research tools 1/Search scientific literature</title>
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      <text bytes="3177" sha1="mp8fidljd9fbop52fx35gmlsu89mgoq" xml:space="preserve">The central theme of research is the research question and to do the literature search related to this question. As written in Google, we build on what others have done &quot;Standing on the shoulders of Giants&quot;. Therefore, every time we start a new project or look for references to solve a specific clinical question, we look in the global scientific health literature. The basic fun search event is defining our search words, those that are most relevant to our questions. By searching and getting some items, we can refine our search words after that. The strategy for search litterature described in an abundance of webpages, You-tubes and articles. One of them in the following:

== [https://www.slideshare.net/sarosem1/medical-literature-search-process Scientific literature search process - open]==

== Search pages ==
The most used search pages are PubMed, Google Scholar, Google Academico and Scielo (See below)  When we use Google Chrome or Firefox as a search engine we can immediately download the results in the reference program, Zotero. The basic fun search event is defining our search words, those that are most relevant to our questions. By searching and getting some items, we can refine our search words after that. It is also important to write in the protocol and later in the article how we have done it. For scientific use, we seek only original articles published in scientific journals, while websites are rarely used. Reading or orienting yourself in scientific articles is necessary to learn how to do this correctly.

== Global search ==
When we conduct our searches, for example, on health and safety awareness among fishermen, we search for international knowledge among fishermen around the world. Now there are no studies on the health and safety of the fishermen of Panama directly and if so, they would also be included in our service when we do not limit our searches to Panama

== Find and store items ==
# Open Zotero/ Mendeley /EndNote and create a file with the name of the project.
# Keep Zotero/ Mendeley open as you search and select interesting articles.
# Click add to store the items in Zotero/ Mendeley
# Moving the file references found in Zotero only works with the &quot;Chrome&quot; and &quot;Firefox&quot; browsers
# Abstracts are always free, but only articles published in Open Access are free. Often summaries are sufficient to obtain relevant information:
====[https://pubmed.ncbi.nlm.nih.gov/ Search PubMed]====

==== [https://scholar.google.dk/ Google Scholar] ====
==== [https://www.cochranelibrary.com/cdsr/reviews Cochrane Library] ====
==== [https://www.dropbox.com/s/8cfd9kuljrtjrhe/Third-edition-How%20to%20search%20valid%20OH%20literature%20English.pdf?dl=0  How to search valid Occupational Health literature] ====
==== [http://repository.icohweb.org/ ICOH Heritage Repository of all Congress proceedings online 1906-2020] ====
==== [https://www.youtube.com/watch?time_continue=181&amp;v=IS07DzG018E How to use Google Scholar] ====
==== [https://www2.le.ac.uk/library/find/databases/p/Prospero Library for Review studies PROSPERO] ====
==== [https://es.cochrane.org/es Cochrane Iberoamérica] ====
==== [https://scielo.org/es/ Scielo] ====</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.   

Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. &lt;nowiki&gt;https://doi.org/10.1186/s13063-021-05500-5&lt;/nowiki&gt;.  

1. Publications and pptx 2016-2026 
== [[/https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e 
/]] ==

https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e 


# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
# Standardized health questionnaires research
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.   

Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. &lt;nowiki&gt;https://doi.org/10.1186/s13063-021-05500-5&lt;/nowiki&gt;.  

1. Publications and pptx 2016-2026 &lt;ref&gt;https://doi.org/10.5603/IMH.2022.0040&lt;/ref&gt;
== [[/https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e 
/]] ==

https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e 


# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
# Standardized health questionnaires research
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.   

Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. &lt;nowiki&gt;https://doi.org/10.1186/s13063-021-05500-5&lt;/nowiki&gt;.  

1. Publications and pptx 2016-2026 &lt;ref&gt;https://doi.org/10.5603/IMH.2022.0040&lt;/ref&gt;


# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
# Standardized health questionnaires research
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.   

Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. &lt;nowiki&gt;https://doi.org/10.1186/s13063-021-05500-5&lt;/nowiki&gt;.  

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;


# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
# Standardized health questionnaires research
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.   

Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''The Institute is a non-profit network of researchers, workers, and students composed of four parts..The Institute is closely related to the Yeung, Kar-Fu, Mihir Gandhi, Amanda Yun Rui Lam, et al. «The Pre-Diabetes Interventions and Continued Tracking to Ease-out Diabetes (Pre-DICTED) program: study protocol for a randomized controlled trial». ''Trials'' 22 (agosto de 2021): 522. &lt;nowiki&gt;https://doi.org/10.1186/s13063-021-05500-5&lt;/nowiki&gt;.  

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;


# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
# Standardized health questionnaires research
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''   

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;
# Screening for Prediabetes, T2 DM, and Hypertension in all health examinations
# Health promotion program integrated with the Prediabetes, T2DM &amp; HTN screening program
# Systematic Literature Reviews
The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.&lt;ref&gt;‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals&lt;/ref&gt;&lt;ref&gt;[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]&lt;/ref&gt; 
We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''&lt;ref&gt; [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]&lt;/ref&gt;,&lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/24737131/&lt;/ref&gt; &lt;ref&gt;https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y&lt;/ref&gt;'''
The research program includes monitoring of the main topics of the EU-Occupational Health strategy 
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[Education 2: Supervision of Students Thesis Projects|Education 2: Supervision of Students' Thesis Projects]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group]] ==
== [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] ==
==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The Health Journal Club]]==
==[[Education 4:Effectiveness of training in prevention for type 2 diabetes|Education 4:  Effectiveness of training in prevention for type 2 diabetes]]==
==[[/Standard Questionnaire Based studies/|Questionnaire Based studies: Protocols and Questionnaires]] ==
==[[/Systematic Reviews/|Systematic Review Studies]]  ==
==[[/Organisation / |Organisation]] ==
==[[/Presentations pptx /|Presentations]] ==
==[[/Invitations for collaboration/]] ==
==[[/DRAFT EU Consortium for Maritime Health Research and Education/|Consortium for Maritime Health Research and Education]]  ==
== Objectives ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group]] ==
# International prospective exposure and health risk cohort studies with maritime students and workers 
# All other industries are invited to promote the similar research and education goals  
# Using the standard protocols with different themes 
# Harmonise exposure and outcome information by using standard questionnaires
# Objective and subjective assessments of workplace hazards exposures 
# Develop and validate Job-exposure Matrices 
# Systematic reviews and pooled studies from the cohort rounds
# Cohort-Linkage to pre-entry and follow-up health exams and other health registers
# Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes  Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension]
# International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing  VCT@WORK''']
# Make training materials based on the cohort study and the clinical study outcomes and other scientific sources.    ....
# OHS training to maritime doctors, seafarers, fishermen, students, and others 
# Integrate research methodology in the supervision of student’s thesis work
# Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data 
# Keep the Excel data file copies safely (producing country and supervisor)
# Disseminate the knowledge in publications and organize seminars/webinars/symposia
==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group/]] ==

==Phases of the preventive program for single industry==
# Epidemiological standardized questionnaire studies 
# Occupational therapists investigate troubled job positions
# Occupational Medical Doctors supply with their patients from the workplace
# Statistics of work accidents in the specific areas of the workplace years
# Dialogue with companies to improve safety and ergonomics positions
# Improve and continue if they have done well

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group/]] ==

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Integrated Mental Health and Work Policy OECD's recommendation ==
To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement;
Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities.
Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace   &lt;ref&gt;https://legalinstruments.oecd.org/public/doc/334/334.en.pdf&lt;/ref&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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== &lt;small&gt;John Snow Research Institute for prediabetes and other metabolic diseases&lt;/small&gt; ==
The Maritime Health research institutes in Esbjerg, Denmark and the international research centre in Cardif, Wales have been closed. The health reseach in the maritime specific areas is now done in other Public Health Institutes.Due to these changes, the objectives in the &quot;Maritime Health Research and Education-NET&quot; has ben transferred to the '''John Snow''' '''Institute''' and objectives has been renewed and now called: '''John Snow Research Institute for prediabetes and other metabolic diseases'''
==[[/|The John Snow International T2 Diabetes Mellitus and Hypertension Research Group/]] ==

==Ethical requirements ==
The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. 
Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect &quot;personal data&quot; as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way
and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version'']
Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials&lt;ref&gt;https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse&lt;/ref&gt;

==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] ==
The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section.

==[[/Contribution to UNs 17 Sustainable Development Goals/]] ==
Goal 3: Good health and well-being for all workers &lt;br&gt;
Goal 4: Quality Education&lt;br&gt;
Goal 5: Gender Equity&lt;br&gt;
Goal 8: Decent Work and Economic Growth&lt;br&gt;
Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)&lt;br&gt;
Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)&lt;br&gt;
Goal 14: Life underwater observations on compliance with good waste management &lt;br&gt;
Goal 17: Partnerships to achieve the Goals&lt;br&gt;

==Contribution to quality education ==
The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases.
===== For the medical doctors doing health examinations=====
Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. &lt;br&gt; https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL
Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases&lt;br&gt;

===== For the students =====
Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data.  They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research.  

==Links to relevant organizations, documents, and funds  ==
[http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health]   
[http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health]   
[[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]]    [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST]   [http://dimopex.eu/about/ DiMoPEx (CA  15129)]   [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications]   [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda]   [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values]   [https://www.europeansurveyresearch.org The European Survey Research Association]   [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)]  [https://ec.europa.eu/esf/main.jsp?catId=67&amp;langId=en&amp;newsId=9691 The European Social Fund] ==
https://www.fi-compass.eu/esif/emff&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] &lt;br&gt;
[https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]&lt;br&gt;
[https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] &lt;br&gt;
[https://www.seafarerstrust.org/about/ ITF Seafarers Trust]&lt;br&gt; 
[https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]
==[https://novonordiskfonden.dk/en/ Novo Nordisk Found]==

Project Title Project Summary Statement of Problem (scientific justification) Justification and Use of Results (final objectives, applicability) Theoretical Framework (argumentation, possible answers, hypothesis) Research Objectives (general and specific) Methodology Type of Study and General Design Operational Definitions (operationalization) Universe of Study, Sample Selection and Size, Unit of Analysis and Observation: Selection and Exclusion Criteria Proposed intervention (if applicable) Data-Collection Procedures, Instruments Used, and Methods for Data Quality Control Procedures to Ensure Ethical Considerations in Research Involving Human Subjects Plan for Analysis of Results Methods and Models of Data Analysis according to Types of Variables Programs to be Used for Data Analysis Bibliographic References Timetable Budget Annexes (data-collection instruments, elaboration on methods and procedures to be used, and more).

==Subpages==
{{Subpages/List}}

'''Bibliographic References.'''

==References==
{{reflist}}
{{CourseCat}}</text>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026
2. Prevalence studies proposals
The-International-Maritime-Health-Database
Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</text>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives</text>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
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    <revision>
      <id>2805908</id>
      <parentid>2805895</parentid>
      <timestamp>2026-04-22T07:20:01Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge. (The Maritime Title is not relevant and asked in the discussion to be changed) 

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
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    <revision>
      <id>2805910</id>
      <parentid>2805908</parentid>
      <timestamp>2026-04-22T07:21:41Z</timestamp>
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        <username>Saltrabook</username>
        <id>1417466</id>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge. (The Maritime Title is not relevant and asked in the discussion to be changed) 

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
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    <revision>
      <id>2805916</id>
      <parentid>2805910</parentid>
      <timestamp>2026-04-22T07:35:08Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
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      <origin>2805916</origin>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge. (The Maritime Title is not relevant and we ask in the discussion to be changed) 

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
      <sha1>4bnxj3xebnhh8l9gf32dlfyiec9e27k</sha1>
    </revision>
    <revision>
      <id>2805917</id>
      <parentid>2805916</parentid>
      <timestamp>2026-04-22T07:37:28Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <origin>2805917</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

==The John Snow Institute==

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge. (Maritime in the title is not relevant and we ask in the discussion to be changed to '''The John Snow Research Institute for prediabetes and other metabolic diseases.''' 

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
      <sha1>g4qvdp0fyp6eiv12y1f37b005ocuu5h</sha1>
    </revision>
    <revision>
      <id>2805919</id>
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www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge. (Maritime in the title is not relevant and we ask in the discussion to be changed to '''The John Snow Research Institute for prediabetes and other metabolic diseases.''' 

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links&lt;/ref&gt;
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives
&lt;references /&gt;</text>
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      <minor />
      <comment>([[c:GR|GR]])  [[File:Acaciafarnesiana1web.jpg]] → [[File:Starr 030202-0020 Acacia farnesiana.jpg]] → File replacement: changing the name of a duplicate file ([[c:c:GR]])</comment>
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      <text bytes="251749" sha1="mh4geq69de4y73rsg5pgz4tvq4d699r" xml:space="preserve">[[Image:Starr 030202-0020 Acacia farnesiana.jpg|thumb|right|250px|Leaves, flowers and fruit of ''Acacia farnesiana'' are shown. Credit: USGS.]]
Medicinal plants are a primary source of organic compounds, both for their medicinal and physiological effects, and for the industrial organic synthesis of a vast array of organic chemicals.&lt;ref&gt;{{cite web|title=Chemicals from Plants|publisher=Cambridge University Botanic Garden
|accessdate=9 December 2017|url=https://web.archive.org/web/20171209152153/http://www.botanic.cam.ac.uk/Botanic/Trail.aspx?p=27&amp;ix=11|archive-date=9 December 2017 }}&lt;/ref&gt; Many hundreds of medicines are derived from plants, both traditional medicines used in herbalism&lt;ref name=&quot;tapsell&quot;&gt;{{cite journal |author1=Tapsell, L.C. |author2=Hemphill, I. |author3=Cobiac, L. |title=Health benefits of herbs and spices: the past, the present, the future |journal=Med. J. Aust. |volume=185 |issue=4 Suppl |pages=S4–24 |date=August 2006 |doi=10.5694/j.1326-5377.2006.tb00548.x |pmid=17022438|s2cid=9769230 |url=https://ro.uow.edu.au/cgi/viewcontent.cgi?article=2450&amp;context=hbspapers }}&lt;/ref&gt;&lt;ref name=&quot;lai&quot;&gt;{{cite journal |author=Lai, P.K.; Roy, J. |title=Antimicrobial and chemopreventive properties of herbs and spices |journal=Curr. Med. Chem. |volume=11 |issue=11 |pages=1451–1460 |date=June 2004 |pmid=15180577 |doi= 10.2174/0929867043365107 |last2=Roy }}&lt;/ref&gt; and chemical substances purified from plants or first identified in them, sometimes by ethnobotanical search, and then organic synthesis for use in modern medicine such as aspirin, taxol, morphine, quinine, reserpine, colchicine, digitalis and vincristine.

Plants used in herbalism include Ginkgo biloba, echinacea, feverfew, and Saint John's wort.

The pharmacopoeia of Dioscorides, ''De Materia Medica'', describing some 600 medicinal plants, was written between 50 and 70 AD and remained in use in Europe and the Middle East until around 1600 AD; it was the precursor of all modern pharmacopoeias.&lt;ref name=NIH&gt;{{cite web
| title=Greek Medicine | publisher=National Institutes of Health, USA | date=16 September 2002 | accessdate=22 May 2014 |url=https://web.archive.org/web/20131109193334/http://www.nlm.nih.gov/hmd/greek/greek_dioscorides.html | archive-date=9 November 2013 }}&lt;/ref&gt;&lt;ref&gt;{{cite book | url=https://books.google.com/books?id=iORoAgAAQBAJ&amp;pg=PA46 | author=Hefferon, Kathleen | title=Let Thy Food Be Thy Medicine | publisher=Oxford University Press | date=2012 |page=46| isbn=978-0199873982 }}&lt;/ref&gt;&lt;ref&gt;{{cite book | url=https://books.google.com/books?id=jBMEAwAAQBAJ&amp;pg=PT143 | author=Rooney, Anne | title=The Story of Medicine | publisher=Arcturus Publishing | date=2009 | page=143| isbn=978-1848580398 }}&lt;/ref&gt;

All plants produce chemical compounds which give them an evolutionary advantage, such as defending against herbivores or, in the example of salicylic acid, as a plant hormone in plant defenses.&lt;ref name=USDAingredients&gt;{{cite web|title=Active Plant Ingredients Used for Medicinal Purposes |url=https://www.fs.fed.us/wildflowers/ethnobotany/medicinal/ingredients.shtml|publisher=United States Department of Agriculture|access-date=18 February 2017 |quote=Below are several examples of active plant ingredients that provide medicinal plant uses for humans.}}&lt;/ref&gt;&lt;ref&gt;{{cite book |title = Salicylic Acid – A Plant Hormone |author1 = Hayat, S. |author2 = Ahmad, A.
|{{isbn|978-1-4020-5183-8}} |date = 2007 |publisher = Springer Science and Business Media
|url = https://archive.org/details/salicylicacidpla0000unse }}&lt;/ref&gt; These phytochemicals have potential for use as drugs, and the content and known pharmacological activity of these substances in medicinal plants is the scientific basis for their use in modern medicine, if scientifically confirmed.&lt;ref name=&quot;ahn&quot;&gt;{{cite journal |last=Ahn |first=K. |title=The worldwide trend of using botanical drugs and strategies for developing global drugs |journal=BMB Reports |year=2017 |volume=50 |issue=3 |pages=111–116 |doi=10.5483/BMBRep.2017.50.3.221 |pmid=27998396 |pmc=5422022 }}&lt;/ref&gt; For instance, daffodils (''Narcissus'') contain nine groups of alkaloids including galantamine, licensed for use against Alzheimer's disease. The alkaloids are bitter-tasting and toxic, and concentrated in the parts of the plant such as the stem most likely to be eaten by herbivores; they may also protect against parasites.&lt;ref&gt;{{cite book |last1=Bastida |first1=Jaume |last2=Lavilla |first2=Rodolfo |last3=Viladomat |first3=Francesc Viladomat |editor1-last=Cordell |editor1-first=G. A. |chapter=Chemical and Biological Aspects of ''Narcissus'' Alkaloids |date=2006 |volume=63 |pages=87–179 |doi=10.1016/S1099-4831(06)63003-4 |pmid=17133715 |title=The Alkaloids: Chemistry and Biology
|isbn=978-0-12-469563-4}}&lt;/ref&gt;&lt;ref&gt;{{cite web |title=Galantamine |url=https://www.drugs.com/mtm/galantamine.html |publisher=Drugs.com |date=2017 |accessdate=17 March 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |pmid=16437532 |date=2006 |last1=Birks |first1=J. |editor1-first=Jacqueline S |editor1-last=Birks |title=Cholinesterase inhibitors for Alzheimer's disease |journal=The Cochrane Database of Systematic Reviews |issue=1 |pages=CD005593 |doi=10.1002/14651858.CD005593}}&lt;/ref&gt;
{{clear}}

==Families==
{{div col|colwidth=20em}}
# ''Actinidiaceae'' - 1
# ''Adoxaceae'' - 6 {{main|Remedy/Plants/Adoxaceae}}
# ''Anacardiaceae'' - 2
# ''Apiaceae'' - 1
# ''Araliaceae'' - 1
# ''Araucariaceae'' - 1
# ''Asphodelaceae'' - 2
# ''Asteraceae'' - 27 {{main|Remedy/Plants/Asteraceae}}
# ''Bangiaceae'' - 5
# ''Berberidaceae'' - 3
# ''Betulaceae'' - 1
# ''Bixaceae'' - 1
# ''Boraginaceae'' - 1
# ''Brassicaceae'' - 1
# ''Burseraceae'' - 2
# ''Cactaceae'' - 1
# ''Caprifoliaceae'' - 2
# ''Cucurbitaceae'' - 1
# ''Elaeagnaceae'' - 1
# ''Equisetaceae'' - 1
# ''Ericaceae'' - 2
# ''Euphorbiaceae'' - 1
# ''Fabaceae'' - 11 {{main|Remedy/Plants/Fabaceae}}
# ''Ginkgoaceae'' - 1
# ''Juglandaceae'' - 5
# ''Lamiaceae'' - 4
# ''Lauraceae'' - 6
# ''Linaceae'' - 2
# ''Lycopodiaceae'' - 1
# ''Lythraceae'' - 1
# ''Magnoliaceae'' - 2
# ''Malvaceae'' - 6
# ''Marchantiaceae'' - 1
# ''Melanthiaceae'' - 1
# ''Meliaceae'' - 1
# ''Menispermaceae'' - 1
# ''Microcoleaceae'' - 1
# ''Moraceae'' - 4
# ''Myristicaceae'' - 3
# ''Myrtaceae'' - 5
# ''Olacaceae'' - 1
# ''Oleaceae'' - 1
# ''Papaveraceae'' - 2
# ''Plantaginaceae'' - 2
# ''Poaceae'' - 5
# ''Ranunculaceae'' - 4
# ''Rosaceae'' - 6 {{main|Remedy/Plants/Rosaceae}}
# ''Rubiaceae'' - 10 {{main|Remedy/Plants/Rubiaceae}}
# ''Rutaceae'' - 4
# ''Simaroubaceae'' - 2
# ''Solanaceae'' - 4
# ''Taxaceae'' - 1
# ''Theaceae'' - 3
# ''Vitaceae'' - 1
# ''Zingiberaceae'' - 5
# ''Zygophyllaceae'' - 1
{{Div col end}}

==''Acacia farnesiana''==
{{main|Remedy/Plants/Fabaceae}}

==''Achillea millefolium''==
{{main|Remedy/Plants/Asteraceae}}

==''Actinidia deliciosa''==
[[Image:Kiwis 006eue.jpg|thumb|right|250px|Kiwi fruit is from the fuzzy kiwifruit. Credit: [[:gl:User:Lmbuga|Luis Miguel Bugallo Sánchez]].{{tlx|free media}}]]
Family: ''Actinidiaceae''.

Kiwifruit seed oil contains on average 62% alpha-linolenic acid, an omega-3 fatty acid.&lt;ref name=Piombo&gt;{{cite journal |url=https://agritrop.cirad.fr/534935/1/document_534935.pdf |last1=Piombo |first1=Georges |last2=Barouh |first2=Nathalie |last3=Barea |first3=Bruno |last4=Renaud |first4=Boulanger |last5=Brat |first5=Pierre |last6=Pina |first6=Michel |last7=Villeneuve |first7=Pierre |title=Characterization of the seed oils from kiwi (Actinidia chinensis), passion fruit (Passiflora edulis) and guava (Psidium guajava) |journal=OCL - Oilseeds and Fats, Crops and Lipids |volume=13 |issue=2 |pages=195–199 |year=2006 |doi=10.1051/ocl.2006.0026 }}&lt;/ref&gt; Kiwifruit pulp contains carotenoids, such as provitamin A beta-carotene,&lt;ref name=Kim&gt;{{cite journal |authors=Kim M, Kim SC, Song KJ, Kim HB, Kim IJ, Song EY, Chun SJ |s2cid=23341156 |title=Transformation of carotenoid biosynthetic genes using a micro-cross section method in kiwifruit (''Actinidia deliciosa'' cv. Hayward) |journal=Plant Cell Reports |volume=29|issue=12|pages=1339–1349|date=Sep 2010 |pmid=20842364 |doi=10.1007/s00299-010-0920-y}}&lt;/ref&gt; lutein and zeaxanthin.&lt;ref name=Sommerburg&gt;{{cite journal |authors=Sommerburg O, Keunen JE, Bird AC, van Kuijk FJ |title=Fruits and vegetables that are sources for lutein and zeaxanthin: the macular pigment in human eyes |journal=British Journal of Ophthalmology |volume=82 |issue=8 |pages=907–910 |date=August 1998 |pmid=9828775 |pmc=1722697 |doi=10.1136/bjo.82.8.907}}&lt;/ref&gt;

{|class=&quot;wikitable&quot;
|-
! Fruit !! Nutritional value !! Second fruit !! Nutritional value 
|-
| Zespri SunGold || kJ=262 || fuzzy fruit || kJ = 255 
|-
| Zespri SunGold || water=82 g || fuzzy fruit || 83 g 
|-
| Zespri SunGold || protein=1.02 g || fuzzy fruit || protein=1.14 h 
|-
| Zespri SunGold || fat=0.28 g || fuzzy fruit || fat=0.52 g
|-
| Zespri SunGold || carbs=15.8 g || fuzzy fruit || carbs=14.66 g
|-
| Zespri SunGold || fiber=1.4 g || fuzzy fruit || fiber=3 g
|-
| Zespri SunGold || sugars=12.3 g || fuzzy fruit || sugars=8.99 g
|-
| Zespri SunGold || calcium_mg=17 || fuzzy fruit || calcium_mg=34
|-
| Zespri SunGold || iron_mg=0.21 || fuzzy fruit || iron_mg=0.31
|-
| Zespri SunGold || magnesium_mg=12 || fuzzy fruit || magnesium_mg=17
|-
| Zespri SunGold || manganese_mg=0.048 || fuzzy fruit || manganese_mg=0.098
|-
| Zespri SunGold || phosphorus_mg=25 || fuzzy fruit || phosphorus_mg=34
|-
| Zespri SunGold || potassium_mg=315 || fuzzy fruit || potassium_mg=312
|-
| Zespri SunGold || sodium_mg=3 || fuzzy fruit || sodium_mg=3
|-
| Zespri SunGold || selenium_ug=0.4 || fuzzy fruit || selenium_ug=0.2
|-
| Zespri SunGold || zinc_mg=0.08 || fuzzy fruit || zinc_mg=0.14
|-
| Zespri SunGold || copper_mg=0.151 || fuzzy fruit || copper_mg=0.13
|-
| Zespri SunGold || vitA_ug= || fuzzy fruit || vitA_IU=
|-
| Zespri SunGold || vitC_mg=161.3 || fuzzy fruit || vitC_mg=92.7
|-
| Zespri SunGold || thiamin_mg=0 || fuzzy fruit || thiamin_mg=0.027
|-
| Zespri SunGold || riboflavin_mg=0.074 || fuzzy fruit || riboflavin_mg=0.025
|-
| Zespri SunGold || niacin_mg=0.231 || fuzzy fruit || niacin_mg=0.341
|-
| Zespri SunGold || pantothenic_mg=0.12 || fuzzy fruit || pantothenic_mg=0.183
|-
| Zespri SunGold || vitB6_mg=0.079 || fuzzy fruit || vitB6_mg=0.063
|-
| Zespri SunGold || folate_ug=31 || fuzzy fruit || folate_ug=25
|-
| Zespri SunGold || vitB12_ug=0.08 || fuzzy fruit || vitB12_ug=0
|-
| Zespri SunGold || choline_mg=1.9 || fuzzy fruit || choline_mg=7.8
|-
| Zespri SunGold || lutein_ug=24 || fuzzy fruit || lutein_ug=122
|-
| Zespri SunGold || vitE_mg=1.4 || fuzzy fruit || vitE_mg=1.46
|-
| Zespri SunGold || vitK_ug=6.1 || fuzzy fruit || vitK_ug=40.3
|-
| sourceusda =[http://ndb.nal.usda.gov/ndb/search/list?qlookup=09520&amp;format=Full link to USDA Database entry]
|}

Raw kiwifruit contains actinidain (also spelled ''actinidin'') which is commercially useful as a meat tenderizer&lt;ref name=Bekhit&gt;{{cite journal|pmid=24499119|year=2014|last1=Bekhit|first1=A. A.|title=Exogenous proteases for meat tenderization|journal=Critical Reviews in Food Science and Nutrition|volume=54|issue=8|pages=1012–31|last2=Hopkins|first2=D. L.|last3=Geesink|first3=G|last4=Bekhit|first4=A. A.|last5=Franks|first5=P|s2cid=57554|doi=10.1080/10408398.2011.623247}}&lt;/ref&gt; and possibly as a digestive aid.&lt;ref name=Boland&gt;{{Cite book|pmid=23394982|year=2013|last1=Boland|first1=M|title=Kiwifruit proteins and enzymes: Actinidin and other significant proteins|journal=Advances in Food and Nutrition Research|volume=68|pages=59–80|doi=10.1016/B978-0-12-394294-4.00004-3|isbn=9780123942944}}&lt;/ref&gt;

==''Adonis vernalis''==
[[Image:Adonis vernalis gonsenheim.jpg|thumb|right|250px|Flowers of ''Adonis vernalis'' are shown. Credit: [[:de:user:Martin Bahmann|Martin Bahmann]].{{tlx|free media}}]]
Family: ''Ranunculaceae''

The plant is poisonous, containing cardiostimulant compounds, such as adonidin and aconitic acid.&lt;ref&gt;{{cite web|url=http://www.henriettesherbal.com/eclectic/kings/adonis.html|title=King's American Dispensatory: ''Adonis''|accessdate=17 April 2006}}&lt;/ref&gt; In addition, it is often used as an ornamental plant.&lt;ref name=Bailey&gt;{{cite book | author=Bailey, L. H. | title=Manual of Gardening (Second Edition)
| url=https://www.gutenberg.org/ebooks/9550 | date=2005 | publisher=Project Gutenberg Literary Archive Foundation }}&lt;/ref&gt; Infusions of the plant are used in the medicine Bekhterev's mixture.&lt;ref&gt;{{cite web|url=https://lekarstvennik.ru/spravochnik-lekarstv/mikstura-bekhtereva|title=Микстура Бехтерева|accessdate=1 April 2018|work=LEKARSTVENNIK.RU}}&lt;/ref&gt;

Due to the cardiac-enhancing effects of Adonis species (including Adonis vernalis), this plant has a history of use in European and Chinese folk medicine.&lt;ref name=Shang/&gt; This plant has been utilized for many different issues and health problems.  The local people of the Soviet Union at one point used it to treat edema or swelling in the body, and an ethanolic extract of the aerial parts of the plant were prepared as an alternative cardiac agent.&lt;ref name=Shang&gt;Shang, Xiaofei; Maio, Xiaolou; Yang, Feng; Wang, Chunmei; Li, Bing; Wang, Weiwei; Pan, Hu; Guo, Xiao; Zhang, Yu; Zhang, Jiyu (4 February 2019). &quot;The Genus Adonis as an Important Cardiac Folk Medicine: A Review of the Ethnobotany, Phytochemistry and Pharmacology&quot;. Frontiers in Pharmacology. 10: 25. doi:10.3389/fphar.2019.00025. {{PMID|30778296}}. Retrieved 22 April 2020.&lt;/ref&gt; In 1879, a Russian medical doctor, N. O. Buhnow, first introduced into medicine alcoholic extracts of the plant as a cardiac stimulant.&lt;ref name=Shikov&gt;Shikov, Alexander N.; Pozharitskaya, Olga N.; Makarov, Valery G.; Wagner, Hildebert; Verpoorte, Rob; Heinrich, Michael (3 July 2014). &quot;Medicinal Plants of the Russian Pharmacopoeia; their history and applications&quot;. Journal of Ethnopharmacology. 153 (3): 481–536. doi:10.1016/j.jep.2014.04.007. Retrieved 22 April 2020.&lt;/ref&gt; In 1898, a mixture of the plant extracts with sodium bromide or codeine was suggested (by Vladimir Bekherev) to treat heart diseases, panic disorder, dystonia and mild forms of epilepsy.&lt;ref name=Shikov&gt;Shikov, Alexander N.; Pozharitskaya, Olga N.; Makarov, Valery G.; Wagner, Hildebert; Verpoorte, Rob; Heinrich, Michael (3 July 2014). &quot;Medicinal Plants of the Russian Pharmacopoeia; their history and applications&quot;. Journal of Ethnopharmacology. 153 (3): 481–536. doi:10.1016/j.jep.2014.04.007. Retrieved 22 April 2020.&lt;/ref&gt; Aqueous infusions of the aerial parts of the plant have been traditionally used in Siberia against edema, cardiac edema and several other issues that are heart related, kidney diseases, and even malaria.&lt;ref name=Shikov/&gt; The biological activity of this extract was defined as 50–66 frog units (amount or liquid of substance that causes the arrest of the heart of a frog) and 6.3–8.0 cat units (amount or liquid of substance that causes the arrest of the heart of a cat) and large enough doses can be toxic.&lt;ref name=Shikov/&gt;

There are many phytochemicals that come from the plant Adonis vernalis and these include cardiac glycosides, other glycosides, and flavones.  The compounds that are cardiac glycosides include Cymarin, Adonitoxin, 16-Hydroxy-strophanthidin, Acetyladonitoxin, Vernadigin, 3-Acetylstrophagogenin, Substance N, Strophanthidine fucoside, 3-Epi-periplogenin, 17β-(2’,5’-dihydro-5’-oxo-3’-furyl)-5β-14β-androstane-3α,5β,14β-triol, Adonitoxigenin 2-O-acetylrhamnosidoxyloside, Adonitoxigenin 3-O-acetylrhamnosidoxyloside, Adonitoxigenin rhamnosidoxyloside, Adonitoxigenin 3-O-[β-D-glucopyranosyl-(1→4)-α-L-rhamnopyranoside, Adonitoxigenin 3-O-[β-D-glucopyranosyl-(1→4)-α-L-(3’-O-acetyl)-rhamnopyranoside, Adonitoxigenin-3-[O-α-L-(2’-O-acetyl) rhamnosido-β-D-glucoside, Digitoxigenin.  Other glycosides include Adonilide, Fukujusonorone, Fukujusone, 12-O-Nicotinoylisolineolon (Lineolon), 12-O-Benzoylisolineolon, Nicotinoylisoramanone, and Isoramanone (digipurprogenin-II).  Flavones include Adonivernith (luteolin-8-hexityl monoxyloside), Homoadonivernith, Orientin, Homoorientin, Isoorientin, Luteolin, and Vitexin.&lt;ref name=Shang/&gt;

The plant contains cardiac glycosides, and these improve the heart's efficiency by increasing its output at the same time as slowing down its rate.&lt;ref&gt;Rouhi, Hossein Reza; Aboutalebian, Mohammad Ali; Saman, Maryam; Karimi, Fatemeh; Champiri, Roya Mahmoudieh (2013). &quot;SEED GERMINATION AND DORMANCY BREAKING METHODS FOR PHEASANT'S EYE (Adonis vernalis L.)&quot;(PDF). International Journal of Agriculture: Research and Review. 3 (1): 172–175. Retrieved 22 April 2020.&lt;/ref&gt;  These glycosides also have a sedative effect and is often prescribed to patients whose hearts are beating irregularly or at an increased rate.&lt;ref&gt;Rouhi, Hossein Reza; Aboutalebian, Mohammad Ali; Saman, Maryam; Karimi, Fatemeh; Champiri, Roya Mahmoudieh (2013). &quot;SEED GERMINATION AND DORMANCY BREAKING METHODS FOR PHEASANT'S EYE (Adonis vernalis L.)&quot;(PDF). International Journal of Agriculture: Research and Review. 3 (1): 172–175. Retrieved 22 April 2020.&lt;/ref&gt; Tinctures of Adonis vernalis are also used by homeopathic physicians in patients that are suffering from congestive cardiac failure and its action is very similar to digitalis (another drug that stimulates the heart muscle).&lt;ref&gt;Esmail, Al-Snafi Ali (2015). &quot;THERAPEUTIC PROPERTIES OF MEDICINAL PLANTS: A REVIEW OF PLANTS WITH CARDIOVASCULAR EFFECTS (PART 1)&quot;. International Journal of Pharmacology &amp; Toxicology. 5 (3): 163–176. Retrieved 22 April 2020.&lt;/ref&gt;  Aqueous extracts of Adonis vernalis were found to have cardiac stimulant effects on isolated heart preparations and it also showed that production of excessive and high potassium concentrations protects against heart failure.&lt;ref&gt;Esmail, Al-Snafi Ali (2015). &quot;THERAPEUTIC PROPERTIES OF MEDICINAL PLANTS: A REVIEW OF PLANTS WITH CARDIOVASCULAR EFFECTS (PART 1)&quot;. International Journal of Pharmacology &amp; Toxicology. 5 (3): 163–176. Retrieved 22 April 2020.&lt;/ref&gt; Not only are cardiac glycosides derived from this plant but there are also some well-known flavones that were identified with pharmacological activities, including antioxidant, antimicrobial, anti-inflammatory, neuro and cardioprotective, and anti-allergic properties.&lt;ref name=Shang/&gt;
{{clear}}

==''Aloe arborescens''==
[[Image:Aloe arborescens Compton.JPG|thumb|right|250px|Jade plants and Krantz aloes in Kirstenbosch Botanical Gardens, Cape Town. Credit: [[c:user:Andrew massyn|Andrew massyn]].{{tlx|free media}}]]
Family: ''Asphodelaceae''.

&quot;Antidiabetic effects of dietary administration of ''Aloe arborescens'' Miller components on multiple lowdose streptozotocin-induced diabetes in mice.&quot;&lt;ref name=Lanka/&gt;
{{clear}}

==''Aloe vera''==
[[Image:Aloe vera flower inset.png|thumb|right|250px|''Aloe vera'' has a flower inset image. Credit: [[w:user:MidgleyDJ]].{{tlx|free media}}]]
Family: ''Asphodelaceae''.

''Aloe vera'' leaves contain phytochemicals under study for possible bioactivity, such as acetylated mannans, polymannans, anthraquinone C-glycosides, anthrones, and other anthraquinones, such as emodin and various lectins.&lt;ref name=&quot;King&quot;&gt;{{cite journal | authors = King GK, Yates KM, Greenlee PG, Pierce KR, Ford CR, McAnalley BH, Tizard IR | title = The effect of Acemannan Immunostimulant in combination with surgery and radiation therapy on spontaneous canine and feline fibrosarcomas | journal = J Am Anim Hosp Assoc | volume = 31 | issue = 5 | pages = 439–447 | year = 1995 | pmid = 8542364 | doi = 10.5326/15473317-31-5-439 }}&lt;/ref&gt;&lt;ref name=&quot;Eshun&quot;&gt;{{cite journal | authors = Eshun K, He Q | title = Aloe vera: a valuable ingredient for the food, pharmaceutical and cosmetic industries—a review | journal = Critical Reviews in Food Science and Nutrition | volume = 44 | issue = 2 | pages = 91–96 | year = 2004 | pmid = 15116756 | doi = 10.1080/10408690490424694 | s2cid = 21241302 }}&lt;/ref&gt;

For people with [[allergies]] to ''Aloe vera'', skin reactions may include contact dermatitis with hives, mild redness and itching, difficulty with breathing, or swelling of the face, lips, tongue, or throat.&lt;ref name=Aloe&gt;{{cite web |title=Aloe |url=https://www.drugs.com/npp/aloe.html |publisher=Drugs.com |accessdate=1 July 2021 |date=30 December 2020}}&lt;/ref&gt;&lt;ref name=AloeVera&gt;{{cite web|url=https://nccih.nih.gov/health/aloevera |title=Aloe vera |publisher=National Center for Complementary and Integrative Health, US National Institutes of Health |date=1 October 2020 |accessdate=1 July 2021}}&lt;/ref&gt;&lt;ref name=&quot;Expert Panel&quot;&gt;{{cite journal | author = Cosmetic Ingredient Review Expert Panel | title = Final Report on the Safety Assessment of Aloe Andongensis Extract, Aloe Andongensis Leaf Juice, Aloe Arborescens Leaf Extract, Aloe Arborescens Leaf Juice, Aloe Arborescens Leaf Protoplasts, Aloe Barbadensis Flower Extract, Aloe Barbadensis Leaf, Aloe Barbadensis Leaf Extract, Aloe Barbadensis Leaf Juice, Aloe Barbadensis Leaf Polysaccharides, Aloe Barbadensis Leaf Water, Aloe Ferox Leaf Extract, Aloe Ferox Leaf Juice, and Aloe Ferox Leaf Juice Extract
|journal = Int. J. Toxicol. | volume = 26 | issue = Suppl 2 | pages = 1–50 | year = 2007 | pmid = 17613130 | doi = 10.1080/10915810701351186 | s2cid = 86018076 | accessdate = 24 May 2016 | archive-date = 15 December 2017 |url = https://web.archive.org/web/20171215084026/http://gov.personalcarecouncil.org/ctfa-static/online/lists/cir-pdfs/pr274.pdf }}&lt;/ref&gt;

&quot;Aloe vera could serve as a natural antihistamine herb. The antihistamine properties of aloe could be attributed, at least in part, to the presence of glycoprotein alprogen which has been demonstrated to antigen-antibody-mediated release of histamine and leukotriene from mast cells (45).&quot;&lt;ref name=Panahi&gt;{{ cite journal
|author=Yunes Panahi, Seyyed Masoud Davoudi, Amirhossein Sahebkar, Fatemeh Beiraghdar, Yahya Dadjo, Iraj Feizi, Golnoush Amirchoopani &amp; Ali Zamani
|title=Efficacy of Aloe vera/olive oil cream versus betamethasone cream for chronic skin lesions following sulfur mustard exposure: a randomized double-blind clinical trial
|journal=Cutaneous and Ocular Toxicology
|date=12 October 2011
|volume=31
|issue=2
|pages=95-103
|url=https://www.tandfonline.com/doi/abs/10.3109/15569527.2011.614669
|arxiv=
|bibcode=
|doi=10.3109/15569527.2011.614669
|pmid=
|accessdate=1 January 2022 }}&lt;/ref&gt;

&quot;Aloe vera contains alprogen as one of the active compound that works by inhibition the absorption of glucose in the digestive tract so that it can reduce blood glucose levels.&quot;&lt;ref name=Iswandi&gt;{{ cite journal
|author=Darwis Iswandi, Graharti Risti, Asthri Agtara Liza
|title=Potency of Aloe vera as Antidiabetic, Antioxidant, and Antilipidemic Therapeutic Modalities
|journal=Potency of Aloe vera as Antidiabetic, Antioxidant, and Antilipidemic Therapeutic Modalities
|date=19 November 2019
|volume=8
|issue=1
|pages=268-272
|url=http://repository.lppm.unila.ac.id/17095/1/Majority%20Maret%2019%20Mahasiswa.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=1 January 2022 }}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot;
|+ Active components present in ''Aloe vera'' with properties&lt;ref name=Lanka&gt;{{ cite journal
|author=Suseela Lanka
|title=A review on ''Aloe Vera'' - the wonder medicinal plant
|journal=Journal of Drug Delivery &amp; Therapeutics
|date=15 October 2018
|volume=8
|issue=5-s
|pages=94-99
|url=http://jddtonline.info/index.php/jddt/article/download/1962/1393
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=1 January 2022 }}&lt;/ref&gt; 
|-
! Name of the Active component !! Active components present in ''Aloe Vera'' with properties 
|-
| Vitamins || Vitamin A (beta-carotene), C and E, - antioxidants. It also contains vitamin B1, B2, B6 &amp; B12, folic acid, and choline.
* Antioxidants protect the body by neutralizing free radicals. 
|-
| Enzymes || Aliiase, alkaline phosphatase, amylase, oxidase, bradykinase, carboxypeptidase, catalase, cellulase, lipase, cylooxygenase, and peroxidase.
*Bradykinase helps to reduce excessive inflammation when applied to the skin topically, while the other enzymes help in the breakdown of sugars, proteins and fats. 
|-
| Minerals || Calcium, chromium, copper, selenium, magnesium, manganese, potassium, sodium and zinc. *Some of the minerals are essential for the proper functioning of various enzyme systems in different metabolic pathways and few acts as antioxidants. 
|-
| Sugars || Monosaccharides (glucose and fructose) and polysaccharides (glucomannans/polymannose). * The most prominent monosaccharide is mannose-6-phosphate, and the most common polysaccharides are called glucomannans [beta-(1,4)-acetylated mannan].
* Acemannan, a prominent glucomannan has also been found. Recently, a glycoprotein with anti allergic properties, called alprogen and novel anti-inflammatory compound, C-glucosyl chromone, has been isolated from Aloe vera gel&lt;sup&gt;15,16&lt;/sup&gt;. 
|-
| Organic acids || Sorbate, salicylic acid, uric acid
* salicylic acid possesses anti-inflammatory and antibacterial properties. 
|-
| Anthraquinones || Aloin, barbaloin, isobarbaloin, anthranol, aloetic acid, aloe-emodin, ester of cinnamic acid, resistannol, chrysophannic acid and emodin,
* Acts as laxatives.
* Aloin and emodin act as analgesics, antibacterials and antivirals. 
|-
| Fatty acids and Steroids || Cholesterol, campesterol, β-sisosterol and lupeol.
Fattyacids like Arachidonic acid, γ-linolenic acid.
* All these have anti-inflammatory action and lupeol also possesses antiseptic and analgesic properties. 
|-
| Non-essential aminoacids || Histidine, arginine, aspartic acid, glutamic acid, proline, glycine, tyrosine, alanine and hydroxyl proline. 
|-
| Essential aminoacids || Methionine, phenylalanine, isoleucine, leucine, valine, threonine and lysine. 
|-
| Hormones || Auxins and gibberellins
* that help in wound healing and have anti-inflammatory action. 
|-
| Others || * Lignin, an inert substance, when included in topical preparations, enhances penetrative effect of the other ingredients into the skin.
* Saponins that are the soapy substances form about 3% of the gel and have cleansing and antiseptic properties. 
|-
|}

&quot;Alprogen, an anti-allergic compound of ''Aloe vera'' inhibits calcium influx into mast cells, thereby inhibiting the antigen-antibody-mediated release of various mediators like histamine, serotonin, SRSA, leukotrienes etc from mast cells&lt;sup&gt;22&lt;/sup&gt;.&quot;&lt;ref name=Lanka/&gt;
{{clear}}

==''Ambrosia acanthicarpa''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia ambrosioides''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia arborescens''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia artemisiifolia''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia confertiflora''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia cordifolia''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia deltoidea''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia dumosa''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia eriocentra''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia ilicifolia''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia monogyra''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia psilostachya''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia salsola''==
{{main|Remedy/Plants/Asteraceae}}

==''Ambrosia trifida''==
{{main|Remedy/Plants/Asteraceae}}

==''Anacardium occidentale''==
[[Image:Cashew apples.jpg|thumb|right|250px|Ripe cashew apples are shown. Credit: [[c:user:Abhishek Jacob|Abhishek Jacob]].{{tlx|free media}}]]
Family:	''Anacardiaceae''.

The cashew tree (''Anacardium occidentale'') is a tropical evergreen tree that produces the cashew seed and the cashew apple accessory fruit.&lt;ref name=Morton&gt;{{cite book
|title=Cashew apple, ''Anacardium occidentale'' L.|publisher=Center for New Crops and Plant Products, Department of Horticulture and Landscape Architecture, Purdue University, W. Lafayette, IN|work=Fruits of warm climates, Julia F. Morton|{{isbn|978-0-9610184-1-2}}|year=1987|author=Morton, Julia F|pages=239–240|accessdate=18 March 2007|url=https://web.archive.org/web/20070315023810/http://www.hort.purdue.edu/newcrop/morton/index.html|archive-date=15 March 2007 }}&lt;/ref&gt;&lt;ref name=&quot;cabi&quot;&gt;{{cite web |title=''Anacardium occidentale'' (cashew nut) |url=https://www.cabi.org/isc/datasheet/5064 |publisher=CABI |accessdate=8 May 2021 |date=20 November 2019 }}&lt;/ref&gt; The tree can grow as high as {{convert|14|m|ft|abbr=on}}, but the dwarf cultivars, growing up to {{convert|6|m|ft|abbr=on}}, prove more profitable, with earlier maturity and greater yields. The cashew seed is commonly considered a snack nut (cashew nut) eaten on its own, used in recipes, or processed into cashew cheese or cashew butter.&lt;ref name=Duke1983&gt;{{cite web |author1=James A Duke |title=''Anacardium occidentale'' L. |url=https://hort.purdue.edu/newcrop/duke_energy/Anacardium_occidentale.html |publisher=Handbook of Energy Crops. (unpublished); In: NewCROP, New Crop Resource Online Program, Center for New Crops and Plant Products, Purdue University |accessdate=10 December 2019 |date=1983}}&lt;/ref&gt;

Raw cashews are 5% water, 30% carbohydrates, 44% fat, and 18% protein (table). In a 100 gram reference amount, raw cashews provide 553 Calories, 67% of the Daily Value (DV) in total fats, 36% DV of protein, 13% DV of dietary fiber and 11% DV of carbohydrates.&lt;ref name=&quot;USDACashew&quot;&gt;{{cite web
|title=Full Report (All Nutrients): 12087, Nuts, cashew nuts, raw, database version SR 27|publisher=Agricultural Research Service – United States Department of Agriculture|accessdate=6 August 2015|date=2015|url=https://web.archive.org/web/20150818022119/http://ndb.nal.usda.gov/ndb/foods/show/3677?fg=&amp;man=&amp;lfacet=&amp;count=&amp;max=&amp;sort=&amp;qlookup=&amp;offset=&amp;format=Full&amp;new=&amp;measureby=|archive-date=18 August 2015 }}&lt;/ref&gt; Cashews are rich sources (20% or more of the DV) of dietary minerals, including particularly copper, manganese, phosphorus, and magnesium (79-110% DV), and of thiamin, vitamin B&lt;sub&gt;6&lt;/sub&gt; and vitamin K (32-37% DV) (table).&lt;ref name=&quot;USDACashew&quot;/&gt; Iron, potassium, zinc, and selenium are present in significant content (14-61% DV) (table).&lt;ref name=&quot;USDACashew&quot;/&gt; Cashews (100 grams, raw) contain {{convert|113|mg}} of beta-sitosterol.&lt;ref name=&quot;USDACashew&quot;/&gt;

Cashew nut oil is a dark yellow oil derived from pressing the cashew nuts (typically from lower value broken chunks created accidentally during processing), and is used for cooking or as a salad dressing, where the highest quality oil is produced from a single cold pressing.&lt;ref&gt;{{cite web|url=http://www.smartkitchen.com/resources/cashew-oil|title=Cashew Oil|publisher=Smart Kitchen|accessdate=15 February 2015}}&lt;/ref&gt;
{{clear}}

==''Arctostaphylos uva-ursi''==
[[Image:Common bearberry (&quot;Kinnikinnick&quot;, Arctostaphylos uva-ursi) - fruits and leaves.JPG|thumb|right|200px|Common bearberry, Kinnikinnick (''Arctostaphylos uva-ursi'') - fruits and leaves, photo was taken in the Selkirk Mountains of northern Idaho. Credit: [[c:user:Jrtayloriv|Jesse Taylor]].{{tlx|free media}}]]
Family:''Ericaceae''.

The plant contains diverse phytochemicals, including ursolic acid, tannic acid, gallic acid, some essential oils and resin, hydroquinones (mainly arbutin, up to 17%), tannins (up to 15%), phenolic glycosides and flavonoids.&lt;ref name=&quot;Pegg&quot;&gt;{{cite journal|authors=Pegg, Ronald B.; Rybarczyk, Anna and Amarowicz, Ryszard
|date=2008|title=Chromatographic separation of tannin fractions from a bearberry leaf (''Arctostaphylos Uva-ursi'' L. Sprengel) extract by Se-HPLC|journal=Polish Journal of Food and Nutrition Sciences|volume=58|issue=4|pages=485–490|doi=10.17221/234/2008-cjfs }}&lt;/ref&gt; ''Arctostaphylos uva-ursi'' leaves contain arbutin,&lt;ref name=&quot;drugs2017&quot;&gt;{{cite web |title=Uva ursi |url=https://www.drugs.com/npc/uva-ursi.html |publisher=Drugs.com |accessdate=27 August 2019 |date=19 July 2017}}&lt;/ref&gt;&lt;ref name=&quot;pubchem2019&quot;&gt;{{cite web |title=Arbutin, CID 440936 |publisher=PubChem, National Library of Medicine, US National Institutes of Health |accessdate=19 November 2019 |date=16 November 2019|url=https://pubchem.ncbi.nlm.nih.gov/compound/440936 }}&lt;/ref&gt; which metabolizes to form hydroquinone, a potential [hepatotoxic (liver toxin).&lt;ref name=pubchem2019/&gt;&lt;ref name=&quot;ijt&quot;&gt;{{cite journal|pmid=24296864|date=2013|last1=De Arriba|first1=S. G|title=Risk assessment of free hydroquinone derived from ''Arctostaphylos Uva-ursi'' folium herbal preparations|journal=International Journal of Toxicology|volume=32|issue=6|pages=442–53|last2=Naser|first2=B|last3=Nolte|first3=K. U|doi=10.1177/1091581813507721 }}&lt;/ref&gt;
{{clear}}

==''Argemone mexicana''==
[[Image:Argemone_mexicana_flower_2.jpg|thumb|right|200px|Flower of Mexican Poppy (''Argemone mexicana'') is featured, an introduced weed on Réunion island. Credit: [[c:user:B.navez|B.navez]].{{tlx|free media}}]]
[[Image:Protopine Structural Formula V1.svg|thumb|right|300px|Diagram illustrates the structure of protopine. Credit: [[c:user:WH23|WH23]]{{tlx|free media}}.]]
''Argemone mexicana'' (Mexican poppy,&lt;ref name=BSBI07&gt;BSBI List 2007 (xls). Botanical Society of Britain and Ireland. Archived from the original (xls) on 2015-06-26. Retrieved 2014-10-17.&lt;/ref&gt; Mexican prickly poppy, flowering thistle,&lt;ref name=&quot;Fuller1986&quot;&gt;{{cite book|author=Thomas C. Fuller|title=Poisonous plants of California|url=https://archive.org/details/bub_gb_0-op0XwlDmQC|accessdate=21 April 2013|year=1986|publisher=University of California Press|isbn=978-0-520-05569-8|pages=[https://archive.org/details/bub_gb_0-op0XwlDmQC/page/n207 201]–}}&lt;/ref&gt;) is of the family ''Papaveraceae''.

Berberine is a found in ''Argemone mexicana'' (prickly poppy).

''Argemone mexicana'' seeds contain 22&amp;ndash;36% of a pale yellow non-edible oil, called ''argemone oil'' or ''katkar oil'', which contains the [[toxic]] alkaloids sanguinarine and dihydrosanguinarine.

Four quaternary isoquinoline alkaloids, dehydrocorydalmine, jatrorrhizine, columbamine, and oxyberberine, have been isolated from the whole plant of ''Argemone mexicana''.&lt;ref&gt;{{cite journal |author1=Singh, S. |author2=Singh, T. D. |author3=Singh, V. P. |author4=Pandey, V. B. | title = Quaternary Alkaloids of ''Argemone mexicana'' | journal = Pharmaceutical Biology | volume = 48 | issue = 2 | pages = 158–160 |date=February 2010 | pmid = 20645832 | doi=10.3109/13880200903062622 }}&lt;/ref&gt; Many other alkaloids such as argemexicaines A and B, coptisine, cryptopine, allocryptopine and chelerythrine have also been found in this plant.&lt;ref name=&quot;pmid12624820&quot;&gt;{{cite journal |author=Chang YC, Hsieh PW, Chang FR, Wu RR, Liaw CC, Lee KH, Wu YC |title=Two new protopines argemexicaines A and B and the anti-HIV alkaloid 6-acetonyldihydrochelerythrine from formosan Argemone mexicana |journal=Planta Medica |volume=69 |issue=2 |pages=148–52 |date=February 2003 |pmid=12624820 |doi=10.1055/s-2003-37710 |url= |issn= }}&lt;/ref&gt;

The seed pods secrete a pale yellow latex when cut open. This argemone resin contains berberine and protopine.
{{clear}}

==''Armoracia rusticana''==
[[Image:Armoracia rusticana.jpg|thumb|right|250px|''Armoracia rusticana'' is in the Botanic Garden, Utrecht, Netherlands. Credit: [[c:user:Pethan|Pethan]].{{tlx|free media}}]]
[[Image:Allyl-isothiocyanate-2D-skeletal.png|thumb|left|250px|Allyl isothiocyanate is the pungent ingredient in fresh horseradish sauce. Credit: [[c:user:Benjah-bmm27|Benjah-bmm27]].{{tlx|free media}}]]
Family:	''Brassicaceae''.

The family ''Brassicaceae'' includes mustard, wasabi, broccoli, cabbage, and radish.

The leaves of the plant are edible, either cooked or raw when young.&lt;ref name=Angier&gt;{{Cite book|last=Angier|first=Bradford|url=https://archive.org/details/fieldguidetoedib00angi/page/104/mode/2up|title=Field Guide to Edible Wild Plants|publisher=Stackpole Books|date=1974|{{isbn|0-8117-0616-8}}|location=Harrisburg, PA|pages=104|oclc=799792 }}&lt;/ref&gt;

Allyl isothiocyanate is an unstable compound, degrading over the course of days at {{convert|37|C}}.&lt;ref name=Kawakishi&gt;{{cite journal | last1 = Ohta | first1 = Yoshio | last2 = Takatani | first2 = Kenichi | last3 = Kawakishi | first3 = Shunro | year = 1995 | title = Decomposition Rate of Allyl Isothiocyanate in Aqueous Solution | journal = Bioscience, Biotechnology, and Biochemistry | volume = 59 | pages = 102–103 | doi = 10.1271/bbb.59.102 }}&lt;/ref&gt;


Horseradish contains volatile oils, notably mustard oil.&lt;ref name=Cole&gt;{{cite journal | last1 = Cole | first1 = Rosemary A. | year = 1976 | title = Isothiocyanates, nitriles and thiocyanates as products of autolysis of glucosinolates in ''Cruciferae'' | journal = Phytochemistry | volume = 15 | issue = 5| pages = 759–762 | doi = 10.1016/S0031-9422(00)94437-6 }}&lt;/ref&gt;
{{clear}}

==''Arnica cordifolia''==
{{main|Remedy/Plants/Asteraceae}}

==''Arnica montana''==
{{main|Remedy/Plants/Asteraceae}}

==''Artemisia abrotanum''==
{{main|Remedy/Plants/Asteraceae}}

==''Arthrospira platensis''==
[[Image:SingleSpirulinaInMicroscope4WEB.jpg|thumb|right|250px|A single ''Arthrospira platensis'' colony is shown. Credit: [[c:user:FarmerOnMars|FarmerOnMars]].{{tlx|free media}}]]
Family:	''Microcoleaceae''

A dietary supplement is made from ''A. platensis'' and ''A. maxima'', known as spirulina.&lt;ref name=Ciferri&gt;{{Cite journal 
| last1 = Ciferri | first1 = O. 
| title = Spirulina, the edible microorganism 
| journal = Microbiological Reviews 
| volume = 47 
| issue = 4 
| pages = 551–578 
| year = 1983 
| pmid = 6420655 
| pmc = 283708
| doi = 10.1128/MMBR.47.4.551-578.1983 }}&lt;/ref&gt;

''Arthrospira'' is very rich in proteins,&lt;ref name=Ciferri/&gt;&lt;ref name=FAO&gt;{{ cite book|author=FAO Report|title=A review on culture, production and use of spirulina as food for humans and feeds for domestic animals and fish|date=2008|publisher=Food and agriculture organization of the united nations|location=Rome }}&lt;/ref&gt; and constitute 53 to 68 percent by dry weight of the contents of the cell.&lt;ref name=Phang&gt;{{cite journal|last1=Phang|first1=S. M.|title=Spirulina cultivation in digested sago starch factory wastewater|journal=Journal of Applied Phycology|date=2000|volume=12|issue=3/5|pages=395–400|doi=10.1023/A:1008157731731|s2cid=20718419}}&lt;/ref&gt;

Its protein harbours all essential amino acids.&lt;ref name=FAO/&gt; ''Arthrospira'' also contain high amounts of polyunsaturated fatty acids (PUFAs), about 1.5–2 percent, and a total lipid content of 5–6 percent.&lt;ref name=FAO/&gt; These PUFAs contain the γ-linolenic acid (GLA), an omega-6 fatty acid.&lt;ref name=Spolaore&gt;{{cite journal|last1=Spolaore|first1=Pauline|s2cid=16896655|display-authors=etal|title=Commercial applications of microalgae|journal=Journal of Bioscience and Bioengineering|date=2006|volume=101|issue=2|pages=87–96|doi=10.1263/jbb.101.87|pmid=16569602}}&lt;/ref&gt; Further contents of ''Arthrospira'' include vitamins, minerals and photosynthetic pigments.&lt;ref name=FAO/&gt;
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==''Astragalus membranaceus''==
{{main|Remedy/Plants/Fabaceae}}

==''Azadirachta indica''==
[[Image:Neem (Azadirachta indica) in Hyderabad W IMG_6976.jpg|thumb|right|250px|Neem (''Azadirachta indica'') is shown in Hyderabad, India. Credit: [[c:user:J.M.Garg|J.M.Garg]].{{tlx|free media}}]]
Family:	''Meliaceae''.

&quot;Nimbidin, Azadirachtin and nimbinin are active compounds present in Neem which are responsible for antibacterial activity.&quot;&lt;ref name=Lakshmi&gt;{{ cite journal
|author=T. Lakshmi, Vidya Krishnan, R Rajendran, and N. Madhusudhanan
|title=''Azadirachta indica'': A herbal panacea in dentistry – An update
|journal=Pharmacognosy Review
|date=June 2015
|volume=9
|issue=17
|pages=41-44
|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4441161/
|arxiv=
|bibcode=
|doi=10.4103/0973-7847.156337
|pmid=26009692
|accessdate=5 February 2022 }}&lt;/ref&gt;

&quot;The phytochemical constituents present in neem are nimbidin, nimbin, nimbolide, Azadirachtin, gallic acid, epicatechin, catechin, and margolone.&quot;&lt;ref name=Lakshmi/&gt;

&quot;From nimbidin other active constituents like nimbin, nimbinin, nimbidinin, nimbolide and nimbidic acid have been isolated which are responsible for its biological activities.[16]&quot;&lt;ref name=Lakshmi/&gt;

&quot;Mucoadhesive dental gel containing Azadirachta indica is found to be beneficial in reducing the plaque index and salivary bacterial count comparatively better than chlorhexidine gluconate mouthwash.[33]&quot;&lt;ref name=Lakshmi/&gt;
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==''Bacopa monnieri''==

Family ''Plantaginaceae''.

''Bacopa monnieri'' is used in Ayurveda (Ayurvedic traditional medicine) to improve memory and to treat various ailments.&lt;ref name=Aguiar2013&gt;{{cite journal |last1=Aguiar |first1=Sebastian |last2=Borowski |first2=Thomas |title=Neuropharmacological review of the nootropic herb Bacopa monnieri |journal=Rejuvenation Research |date=2013 |volume=16 |issue=4 |pages=313–326 |doi=10.1089/rej.2013.1431 |pmid=23772955 |pmc=3746283 |issn=1557-8577}}&lt;/ref&gt; Reviews of preliminary research found that ''Bacopa monnieri'' may nootropic (improve cognition),&lt;ref name=Aguiar2013/&gt;&lt;ref name=Konkeaw2014&gt;{{cite journal |last1=Kongkeaw |first1=C |last2=Dilokthornsakul |first2=P |last3=Thanarangsarit |first3=P |last4=Limpeanchob |first4=N |last5=Norman Scholfield |first5=C |title=Meta-analysis of randomized controlled trials on cognitive effects of ''Bacopa monnieri'' extract. |journal=Journal of Ethnopharmacology |date=2014 |volume=151 |issue=1 |pages=528–35 |doi=10.1016/j.jep.2013.11.008 |pmid=24252493}}&lt;/ref&gt; although the effect was measurable only after several weeks of use.&lt;ref name=&quot;Neale&quot;&gt;{{cite journal | last1=Neale | first1=Chris | last2=Camfield | first2=David | last3=Reay | first3=Jonathon | last4=Stough | first4=Con | last5=Scholey | first5=Andrew | title=Cognitive effects of two nutraceuticals Ginseng and Bacopa benchmarked against modafinil: a review and comparison of effect sizes | journal=British Journal of Clinical Pharmacology | volume=75 | issue=3 | date=5 February 2013 | issn=0306-5251 | pmid=23043278 | pmc=3575939 | doi=10.1111/bcp.12002 | pages=728–737}}&lt;/ref&gt;

In 2019, the FDA issued warning letters to manufacturers of dietary supplements containing ''Bacopa monnieri'' that advertised health claims for treating or preventing stomach disease, Alzheimer's disease, hypoglycemia, blood pressure, and anxiety were unproven and illegal. The FDA stated that ''Bacopa monnieri'' products have not been approved for these or any medical purposes.&lt;ref name=&quot;fda-fraud&quot;&gt;{{cite web |title=Health fraud scams: Unproven Alzheimer's disease products (''Bacopa monnieri'' listed)|url=https://www.fda.gov/consumers/health-fraud-scams/unproven-alzheimers-disease-products |publisher=US Food and Drug Administration |accessdate=11 May 2019 |date=22 December 2018}}&lt;/ref&gt;&lt;ref name=&quot;fda&quot;&gt;{{cite web |author1=William A Correll, Jr. |title=FDA Warning Letter: Peak Nootropics LLC aka Advanced Nootropics |url=https://www.fda.gov/inspections-compliance-enforcement-and-criminal-investigations/warning-letters/peak-nootropics-llc-aka-advanced-nootropics-565256-02052019 |publisher=Office of Compliance, Center for Food Safety and Applied Nutrition, Inspections, Compliance, Enforcement, and Criminal Investigations, US Food and Drug Administration |accessdate=11 May 2019 |date=5 February 2019}}&lt;/ref&gt;&lt;ref name=&quot;fda2&quot;&gt;{{cite web |author1=William A Correll, Jr. |title=FDA Warning Letter: TEK Naturals |url=https://www.fda.gov/inspections-compliance-enforcement-and-criminal-investigations/warning-letters/tek-naturals-565026-02042019 |publisher=Office of Compliance, Center for Food Safety and Applied Nutrition, Inspections, Compliance, Enforcement, and Criminal Investigations, US Food and Drug Administration |accessdate=11 May 2019 |date=5 February 2019}}&lt;/ref&gt;

The most commonly reported adverse effects of ''Bacopa monnieri'' in humans are nausea, increased intestinal motility, and gastrointestinal upset.&lt;ref name=Aguiar2013/&gt;

The best characterized phytochemicals  in ''Bacopa monnieri'' are dammarane-type triterpenoid saponins known as bacosides, with jujubogenin or pseudo-jujubogenin moieties as aglycone units.&lt;ref&gt;{{cite journal | last1 = Sivaramakrishna | first1 = C | last2 = Rao | first2 = CV | last3 = Trimurtulu | first3 = G | last4 = Vanisree | first4 = M | last5 = Subbaraju | first5 = GV | date = 2005 | title = Triterpenoid glycosides from ''Bacopa monnieri'' | journal = Phytochemistry | volume = 66 | issue = 23| pages = 2719–2728 | doi=10.1016/j.phytochem.2005.09.016| pmid = 16293276 }}&lt;/ref&gt; Bacosides comprise a family of 12 known analogs.&lt;ref&gt;{{cite journal | last1 = Garai | first1 = S | last2 = Mahato | first2 = SB | last3 = Ohtani | first3 = K | last4 = Yamasaki | first4 = K | date = 2009 | title = Dammarane triterpenoid saponins from ''Bacopa monnieri''| journal = Can J Chem | volume = 87 | issue = 9| pages = 1230–1234|doi=10.1139/V09-111 }}&lt;/ref&gt; Other saponins called bacopasides I–XII were identified.&lt;ref&gt;{{cite journal | last1 = Chakravarty | first1 = A.K | last2 = Garai | first2 = S. | last3 = Masuda | first3 = K | last4 = Nakane | first4 = T | last5 = Kawahara | first5 = N. | date = 2003 | title = Bacopasides III–V: Three new triterpenoid glycosides from ''Bacopa monnieri''| journal = Chem Pharm Bull | volume = 51 | issue = 2| pages = 215–217 | doi=10.1248/cpb.51.215 | pmid=12576661| doi-access = free }}&lt;/ref&gt; The alkaloids brahmine, nicotine, and herpestine have been catalogued, along with D-mannitol, apigenin, hersaponin, monnierasides I–III, cucurbitacin and plantainoside B.&lt;ref&gt;{{cite journal | last1 = Chatterji | first1 = N | last2 = Rastogi | first2 = RP | last3 = Dhar | first3 = ML | date = 1965 | title = Chemical examination of ''Bacopa monniera'' Wettst: Part II—Isolation of chemical constituents | journal = Ind J Chem | volume = 3 | pages = 24–29 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal | last1 = Chakravarty | first1 = AK | last2 = Sarkar | first2 = T | last3 = Nakane | first3 = T | last4 = Kawahara | first4 = N | last5 = Masuda | first5 = K | date = 2008 | title = New phenylethanoid glycosides from ''Bacopa monnieri'' | journal = Chem Pharm Bull | volume = 50 | issue = 12 |doi=10.1248/cpb.50.1616| pmid = 12499603 | pages = 1616–1618 | doi-access = free }}&lt;/ref&gt;&lt;ref name=&quot;Bhandari&quot;&gt;{{cite journal | last1=Bhandari | first1=Pamita | last2=Kumar | first2=Neeraj | last3=Singh | first3=Bikram | last4=Kaul | first4=Vijay K. | title=Cucurbitacins from ''Bacopa monnieri'' | journal=Phytochemistry | volume=68 | issue=9 | date=2007 | issn=0031-9422 | doi=10.1016/j.phytochem.2007.03.013 | pages=1248–1254| pmid=17442350 }}&lt;/ref&gt;

==''Berberis aristata''==
[[Image:Berberin.svg|thumb|right|300px|Chemical structure of berberine, an alkaloid found in ''B. aristata'', is illustrated. Credit: [[c:user:NEUROtiker|NEUROtiker]].{{tlx|free media}}]]
Family ''Berberidaceae''.

Berberine is a quaternary ammonium salt from the protoberberine group of benzylisoquinoline alkaloids found in ''Berberis aristata'' (tree turmeric).&lt;ref&gt;{{cite journal | author = Zhang Q, Cai L, Zhong G, Luo W | title = Simultaneous determination of jatrorrhizine, palmatine, berberine, and obacunone in Phellodendri Amurensis Cortex by RP-HPLC | journal = Zhongguo Zhong Yao Za Zhi = Zhongguo Zhongyao Zazhi = China Journal of Chinese Materia Medica | volume = 35 | issue = 16 | pages = 2061–4 | date = 2010 | pmid = 21046728 | doi = 10.4268/cjcmm20101603 }}&lt;/ref&gt;

The root bark contains the bitter alkaloid berberine, which has been studied for its potential pharmacological properties.&lt;ref&gt;{{cite web | url = http://www.webmd.com/vitamins-supplements/ingredientmono-1126-BERBERINE.aspx | title = Berberine | publisher = WebMD}}&lt;/ref&gt;
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==''Berberis vulgaris''==
[[Image:Berberis vulgaris .jpg|thumb|right|200px|''Berberis vulgaris'' (European barberry)/(Jaundice berry)/(Ambarbaris)/(Barberry) is a shrub in the family Berberidaceae, native to central and southern Europe, northwest Africa and western Asia. Fruit are shown. Credit: [[c:user=Eaglestein|Arnstein Rønning]].{{tlx|free media}}]]
The dried fruit of ''Berberis vulgaris'' (barberry) is used in herbal medicine.&lt;ref&gt;See e.g. [https://web.archive.org/web/20130801084909/http://umm.edu/health/medical/altmed/herb/barberry &quot;Barberry&quot; @ Alternative Medicine @ University of Maryland Medical Center]&lt;/ref&gt; The chemical constituents include isoquinolone alkaloids, especially berberine, with a full list of phytochemicals compiled.&lt;ref&gt;Mokhber-Dezfuli N, Saeidnia S, Gohari AR, Kurepaz-Mahmoodabadi M. Phytochemistry and pharmacology of berberis species. Pharmacogn Rev. 2014;8(15):8–15. doi:10.4103/0973-7847.125517&lt;/ref&gt;
{{clear}}

==''Bixa orellana''==
[[Image:bixin.png|thumb|left|250px|Bixin is the major apocarotenoid of annatto. Credit: [[c:user:Edgar181|Edgar181]].{{tlx|free media}}]]
[[Image:Bixa.jpg|thumb|right|250px|Achiote flower and buds are shown in Lavras, Minas Gerais, Brazil. Credit: Denis Conrado.{{tlx|free media}}]]
Family:	''Bixaceae''.

Bixin is the major apocarotenoid of annatto&lt;ref name=&quot;ntp&quot;&gt;{{cite web
|title=Executive Summary Bixin |work= National Institute of Environmental Health Sciences |publisher=National Institutes of Health |date=November 1997 |accessdate=24 August 2011|url= https://web.archive.org/web/20110721055506/http://ntp-server.niehs.nih.gov/?objectid=F59ACAC5-F1F6-975E-7C563568F5F7351B| archive-date= 21 July 2011 }}&lt;/ref&gt;

The yellow to orange color is produced by the chemical compounds bixin (orange) and norbixin (yellow), which are classified as carotenoids, where the fat-soluble color in the crude extract is called bixin, which can then be saponified into water-soluble norbixin, with the dual solubility property of annatto being rare for carotenoids.&lt;ref name=Smith&gt;{{cite web |last1= Smith |first1= James |first2= Harriet |last2=Wallin |title= Annatto Extracts: Chemical and Technical Assessment |publisher= JECFA (FAO) |date= 2006 |url= http://www.fao.org/fileadmin/templates/agns/pdf/jecfa/cta/67/annatto.pdf |accessdate= 10 June 2013 }}&lt;/ref&gt; The seeds contain 4.5–5.5% pigment, which consists of 70–80% bixin.&lt;ref name=&quot;ntp&quot; /&gt; Unlike beta-carotene, another well-known carotenoid, annatto-based pigments are not vitamin A precursors.&lt;ref name=Kuntz&gt;{{cite web |last= Kuntz |first= Lynn A. |title= Natural Colors: A Shade More Healthy |work= Food Product Design |publisher= Virgo Publishing, LLC |date= 4 August 2008 |url= http://www.foodproductdesign.com/articles/2008/08/natural-colors-a-shade-more-healthy.aspx |access-date= 26 January 2013}}&lt;/ref&gt;

Annatto oil is also rich in tocotrienols, beta-carotene, essential oils, saturated and unsaturated fatty acids, flavonoids, and vitamin C.&lt;ref name=Meireles&gt;{{cite web |last1= Ângela de Almeida Meireles |first1= Maria |first2= Carolina |last2= Lima Cavalcante de Albuquerque |title= Processo otimizado para obtenção de óleo rico em antioxidantes de urucum |work= Inova |publisher= Unicamp |url= http://www.inova.unicamp.br/sici/visoes/ajax/ax_pdf_divulgacao.php?token=QyjKaJjh |accessdate= 2 June 2015 }}&lt;/ref&gt;

''Bixa orellana'' is used in traditional medicine.&lt;ref name=Morton1960&gt;{{cite journal |first=Julia F. |last=Morton |title=Can Annatto (''Bixa orellana'', L.), an old source of food color, meet new needs for safe dye? |url=http://journals.fcla.edu/fshs/article/view/101136/97080 |journal=Proceedings of the Florida State Horticultural Society |volume=73 |pages=301–309 |accessdate=10 October 2018 |year=1960}}&lt;/ref&gt;&lt;ref name=Vilar&gt;{{Cite journal |last1=Vilar |first1=Daniela de Araújo |last2=Vilar |first2=Marina Suênia de Araujo |last3=Moura |first3=Túlio Flávio Accioly de Lima e |last4=Raffin |first4=Fernanda Nervo |last5=Oliveira |first5=Márcia Rosa de |last6=Franco |first6=Camilo Flamarion de Oliveira |last7=de Athayde-Filho |first7=Petrônio Filgueiras |last8=Diniz |first8=Margareth de Fátima Formiga Melo |last9=Barbosa-Filho |first9=José Maria |date=2014 |title=Traditional Uses, Chemical Constituents, and Biological Activities of Bixa orellana L.: A Review |journal=The Scientific World Journal |lang=en |volume=2014 |pages=857292 |doi=10.1155/2014/857292 |issn=2356-6140 |pmc=4094728 |pmid=25050404|doi-access=free }}&lt;/ref&gt; The tree has been used in Ayurveda, the folk medicine practices of India, where different parts of the plant are thought to be useful as therapy.&lt;ref name=Khare&gt;{{cite book |url=https://www.springer.com/medicine/complementary+%26+alternative+medicine/book/978-0-387-70637-5 |title=Indian Medicinal Plants |last=Khare |first=C. P. |date=2007 |publisher=Springer Science+Business Media, LLC |{{isbn|978-0-387-70638-2}} |location=New York}}&lt;/ref&gt;
{{clear}}

==''Boswellia sacra''==
[[Image:Monoterpenes from the essential oil of the Boswellia sacra resin.png|thumb|right|300px|Monoterpenes found in the essential oil of ''Boswellia sacra'' resin. Credit: Ahmed Al-Harrasi and Salim Al-Saidi.{{tlx|fairuse}}]]
[[Image:Boswellia sacra.jpg|thumb|right|200px|''Boswellia sacra'' is shown at Florida International University campus, Miami, Florida, USA. Credit: [https://www.flickr.com/people/12017190@N06 Scott Zona from USA].{{tlx|Free media}}]]
[[Image:Sesquiterpenes from the essential oil of the Boswellia sacra resin.png|thumb|left|300px|Sesquiterpenes found in the essential oil of the ''Boswellia sacra'' resin. Credit: Ahmed Al-Harrasi and Salim Al-Saidi.{{tlx|fairuse}}]]
Family: ''Burseraceae''.

''Boswellia sacra'' (commonly known as frankincense or olibanum-tree)&lt;ref name=grinBoswellia&gt;&quot;Boswellia sacra&quot;. Germplasm Resources Information Network (GRIN). Agricultural Research Service (ARS), United States Department of Agriculture (USDA). Retrieved November 24, 2012.&lt;/ref&gt; is a tree, the primary tree in the genus ''Boswellia'' from which frankincense, a resinous dried sap, is harvested and is native to the Arabian Peninsula (Oman, Yemen), and horn of Africa (Somalia).&lt;ref name=grinBoswellia/&gt;

&quot;The [essential] oil of [''Boswellia sacra''] contains a high proportion of monoterpenes (97.3%) in which ''E''-β-ocimene and limonene were the major constituents. The remaining 2.7% was accounted for by sesquiterpenes, in which ''E''-caryophyllene was the major constituent.&quot;&lt;ref name=Harrasi&gt;{{ cite journal
|author=Ahmed Al-Harrasi and Salim Al-Saidi
|title=Phytochemical analysis of the essential oil from botanically certified oleogum resin of Boswellia sacra (Omani Luban)
|journal=Molecules
|date=27 August 2008
|volume=2008
|issue=13
|pages=2181-2189
|url=https://www.mdpi.com/1420-3049/13/9/2181/pdf
|arxiv=
|bibcode=
|doi=10.3390/molecules13092181
|pmid=
|accessdate=2 September 2021 }}&lt;/ref&gt;

&quot;The monoterpenes were identified as 2-β-pinene (0.1%), α-thujene (6.6%), E-β-ocimene (32.3%), 2,4(10)-thujadiene (0.2%), camphene (0.6%), sabinene (5.2%), 1-β-pinene (1.8%), myrcene (6.9%), α- pinene (5.3%), 2-carene (0.8%), limonene (33.5%), Z-β-ocimene (0.2%), γ-terpinene (1.0%), terpinolene (0.4%), p-cymene (0.2%), 1,4-cyclohexadiene (0.1%), perillene (0.1%), isopentyl-2- methyl butanoate (0.1%), isomyl valerate (0.1%), 1,3,6-trimethylenecycloheptane (0.1%), β-thujone (0.1%), α-campholene aldehyde (0.2%), allo-ocimene (0.1%), trans-pinocarveol (0.1%), p-mentha- 1,5-dien-8-ol (0.2 %), 4-terpineol (0.2%), sabinyl acetate (0.1%), myrtenal (0.1%), α-terpineol (0.1%), α-phellandrene epoxide (0.1%), verbenone (0.1%), trans-(+)-carveol (0.1%), carvone (0.1%) and 1- bornyl acetate (0.1%).&quot;&lt;ref name=Harrasi/&gt; See the image on the right.

&quot;The sesquiterpenes were identified to be α-cubebene (0.1%), α-copaene (0.3%), β-bourbonene (0.1%), β-elemene (0.3%), α-gurjunene (0.1%), E-caryophyllene (0.9%), α-humulene (0.2%), allo-aromadendrene (0.0.1%), α-amorphene (0.1%), germacrene D (0.1%), β-selinene (0.1%), α-selinene (0.1%), α-muurolene (0.1%), γ-cadinene (0.1%), caryophyllene oxide (0.01%) and γ-muurolene (0.1%).&quot;&lt;ref name=Harrasi/&gt; See the image on the left.
{{clear}}

==''Boswellia serrata''==
[[Image:Boswellia serrata (4399783209).jpg|thumb|right|250px|''Boswellia serrata'' flowers are shown. Credit: [https://www.flickr.com/people/91314344@N00 Dinesh Valke from Thane, India].{{tlx|free media}}]]
''Boswellia serrata'' is a plant that produces Indian frankincense, also known as Indian oli-banum, Salai guggul, and Sallaki in Sanskrit.&lt;ref&gt;Pole, Sebastian (2013) Ayurvedic Medicine: The Principles of Traditional Practice. Singing Dragon Press. p.179&lt;/ref&gt; The plant is native to much of India and the Punjab region that extends into Pakistan.&lt;ref&gt;&quot;Boswellia serrata&quot;. Germplasm Resources Information Network (GRIN). Agricultural Research Service (ARS), United States Department of Agriculture (USDA). Retrieved 15 October 2014.&lt;/ref&gt;

''Boswellia serrata'' contains various derivatives of boswellic acid including β-boswellic acid, acetyl-β-boswellic acid, 11-keto-β-boswellic acid and acetyl-11-keto-β-boswellic acid [AKBA].&lt;ref&gt;{{Cite journal|last=Dragos|first=Dorin|last2=Gilca|first2=Marilena|last3=Gaman|first3=Laura|last4=Vlad|first4=Adelina|last5=Iosif|first5=Liviu|last6=Stoian|first6=Irina|last7=Lupescu|first7=Olivera|date=2017-01-16|title=Phytomedicine in Joint Disorders|journal=Nutrients|volume=9|issue=1|pages=70|doi=10.3390/nu9010070|issn=2072-6643|pmc=5295114|pmid=28275210}}&lt;/ref&gt;

Extracts of ''Boswellia serrata'' have been clinically studied for osteoarthritis and joint function, with the research showing trends of benefit (slight improvement) in pain and function.&lt;ref name=&quot;Cameron&quot;&gt;{{cite journal | last=Cameron | first=M | last2=Chrubasik | first2=S | title=Oral herbal therapies for treating osteoarthritis | journal=The Cochrane Database of Systematic Reviews | issue=5 | date=22 May 2014 | issn=1469-493X | pmid=24848732 | pmc=4494689 | doi=10.1002/14651858.CD002947.pub2 | page=CD002947}}&lt;/ref&gt; It has been used in Indian traditional medicine for diabetes.&lt;ref&gt;{{cite journal|title=The Effects of Boswellia serrata Gum Resin on the Blood Glucose and Lipid Profile of Diabetic Patients: A Double-Blind Randomized Placebo-Controlled Clinical Trial.|journal=Journal of Evidence-Based Integrative Medicine|volume=23|pages=2515690X18772728|doi=10.1177/2515690X18772728|pmid=29774768|pmc=5960856|date=2018|last1=Mehrzadi|first1=S.|last2=Tavakolifar|first2=B.|last3=Huseini|first3=H. F.|last4=Mosavat|first4=S. H.|last5=Heydari|first5=M.}}&lt;/ref&gt;

Boswellic acid (acetyl-keto-beta-boswellic acid (AKBA)), one of the bioactive boswellic acids found in ''Boswellia serrata'' (Indian Frankincense) has been found to inhibit 5-lipoxygenase strongly as an allosteric inhibitor.&lt;ref name=Gilbert&gt;{{Cite journal|last=Gilbert|first=Nathaniel C.|last2=Gerstmeier|first2=Jana|last3=Schexnaydre|first3=Erin E.|last4=Börner|first4=Friedemann|last5=Garscha|first5=Ulrike|last6=Neau|first6=David B.|last7=Werz|first7=Oliver|last8=Newcomer|first8=Marcia E.|date=July 2020|title=Structural and mechanistic insights into 5-lipoxygenase inhibition by natural products|url= |journal=Nature Chemical Biology
|volume=16|issue=7|pages=783–790|doi=10.1038/s41589-020-0544-7|issn=1552-4450|pmc=7747934|pmid=32393899}}&lt;/ref&gt; Boswellia administration has been shown to reduce brain edema in patients irradiated for brain tumor and it's believed to be due to 5-lipoxygenase inhibition.&lt;ref name=&quot;Simon Kirste 2009&quot;&gt;{{ cite journal| author=Simon Kirste  | title=Antiödematöse Wirkung von Boswellia serrata auf dasStrahlentherapie - assoziierte Hirnödem | year= 2009 |  url=https://d-nb.info/101045384X/34 }}&lt;/ref&gt;&lt;ref name=&quot;pmid21287538&quot;&gt;{{cite journal| author=Kirste S, Treier M, Wehrle SJ, Becker G, Abdel-Tawab M, Gerbeth K
| title=Boswellia serrata acts on cerebral edema in patients irradiated for brain tumors: a prospective, randomized, placebo-controlled, double-blind pilot trial. | journal=Cancer | year= 2011 | volume= 117 | issue= 16 | pages= 3788–95 | pmid=21287538 | doi=10.1002/cncr.25945 | url=https://www.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?dbfrom=pubmed&amp;tool=sumsearch.org/cite&amp;retmode=ref&amp;cmd=prlinks&amp;id=21287538 }}&lt;/ref&gt;
{{clear}}

==''Calendula officinalis''==
{{main|Remedy/Plants/Asteraceae}}

==''Camellia reticulata''==
[[Image:Camellia reticulata RHS.jpeg|thumb|right|200px|''Camellia reticulata'' is shown in a hand-coloured engraving after a drawing by Alfred Chandler (1804-1896). Credit: [[c:user:BernardM|BernardM]].{{tlx|free media}}]]
Family Theaceae.

''Camellia reticulata'' has a long history of cultivation, both for tea oil and for its ornamental value.&lt;ref&gt;{{cite web |title=Camellias from China |date=8 Mar 2012 |work=Rhododendron Dell &amp;mdash; Plant collections |publisher=Dunedin Botanic Garden |accessdate=5 April 2016 |url=https://web.archive.org/web/20160422203844/http://www.dunedinbotanicgarden.co.nz/collections/plant-life-article/camellias-from-china |archive-date=22 April 2016 }}&lt;/ref&gt;
{{clear}}

==''Camellia sasanqua''==
[[Image:Camellia sasanqua1JAM343.jpg|thumb|right|200px|''Camellia sasanqua'' is used as a garden plant, its leaves are used for tea, and its seeds for oil. Credit: [https://www.flickr.com/people/24841050@N00 junichiro aoyama from Kyoto, Japan].{{tlx|Free media}}]]
Family Theaceae.

The leaves are used to make tea while the seeds or nuts are used to make tea seed oil,&lt;ref&gt;[http://www.botanical-dermatology-database.info/BotDermFolder/THEA.html#Camellia%20sasanqua ''Camellia sasanqua''] in BoDD – Botanical Dermatology Database&lt;/ref&gt; which is used for lighting, lubrication, cooking and cosmetic purposes.
{{clear}}

==''Camellia sinensis''==
[[Image:Teestrauch Detail.jpg|thumb|right|250px|Tea plant is shown in a tea plantation. Credit: [[:de:user:Sebastianjude|Sebastianjude]].{{tlx|Free media}}]]
Family ''Theaceae''.

Polyphenols found in green tea include epigallocatechin gallate (EGCG), epicatechin gallate, epicatechins and flavanols,&lt;ref name=Khan2013&gt;{{cite journal|author=Khan N, Mukhtar H |title=Tea and health: studies in humans|journal=Current Pharmaceutical Design|volume=19|issue=34|pages=6141–7|date=2013|pmid= 23448443|pmc=4055352|type=Literature Review|doi=10.2174/1381612811319340008}}&lt;/ref&gt; which are under laboratory research for their potential effects ''in vivo''.&lt;ref name=Johnson/&gt; Other components include three kinds of flavonoids, known as kaempferol, quercetin, and myricetin.&lt;ref&gt;Committee on Diet, Nutrition, and Cancer, Assembly of Life Sciences, National Research Council, Diet, nutrition, and cancer, Washington: D.C National Academies Press, 1982, p. 286.&lt;/ref&gt; Although the mean content of flavonoids and catechins in a cup of green tea is higher than that in the same volume of other food and drink items that are traditionally considered to promote health,&lt;ref name=&quot;USDA_2007&quot;&gt;USDA Database for the Flavonoid Content of Selected Foods, Release 2.1 (2007)&lt;/ref&gt; flavonoids and catechins have no proven biological effect in humans.&lt;ref name=&quot;efsa&quot;&gt;{{cite web|url=http://www.efsa.europa.eu/en/efsajournal/pub/2055|publisher=European Food Safety Authority|title=Scientific Opinion on the substantiation of health claims related to Camellia sinensis (L.) Kuntze (tea), including catechins in green tea, and improvement of endothelium-dependent vasodilation (ID 1106, 1310), maintenance of normal blood pressure (ID 1310, 2657), maintenance of normal blood glucose concentrations (ID 1108), maintenance of normal blood LDL cholesterol concentrations (ID 2640), protection of the skin from UV-induced (including photo-oxidative) damage (ID 1110, 1119), protection of DNA from oxidative damage (ID 1120, 1121), protection of lipids from oxidative damage (ID 1275), contribution to normal cognitive function (ID 1117, 2812), &quot;cardiovascular system&quot; (ID 2814), &quot;invigoration of the body&quot; (ID 1274, 3280), decreasing potentially pathogenic gastro-intestinal microorganisms (ID 1118), &quot;immune health&quot; (ID 1273) and &quot;mouth&quot; (ID 2813) pursuant to Article 13(1) of Regulation (EC) No 1924/2006|date=8 April 2011|accessdate=9 November 2014}}&lt;/ref&gt;&lt;ref name=&quot;EFSA Panel on Dietetic Products, Nutrition and Allergies NDA2, 3 European Food Safety Authority EFSA, Parma, Italy 2010 1489&quot;&gt;{{cite journal|author=EFSA Panel on Dietetic Products, Nutrition and Allergies (NDA)2, 3 European Food Safety Authority (EFSA), Parma, Italy|title=Scientific Opinion on the substantiation of health claims related to various food(s)/food constituent(s) and protection of cells from premature aging, antioxidant activity, antioxidant content and antioxidant properties, and protection of DNA, proteins and lipids from oxidative damage pursuant to Article 13(1) of Regulation (EC) No 1924/20061|journal= EFSA Journal|date= 2010|volume= 8|issue=2|page=1489|doi=10.2903/j.efsa.2010.1489|doi-access=free}}&lt;/ref&gt;

Green tea leaves are initially processed by soaking in an alcohol solution, which may be further concentrated to various levels; byproducts of the process are also packaged and used. Extracts are sold over the counter in liquid, powder, capsule, and tablet forms,&lt;ref name=Johnson&gt;I.T. Johnson &amp; G. Williamson, Phytochemical functional foods, Cambridge, UK: Woodhead Publishing, 2003, pp. 135-145&lt;/ref&gt;&lt;ref&gt;A. Bascom, Incorporating herbal medicine into clinical practice, Philadelphia: F.A. Davis Company, 2002, p. 153.&lt;/ref&gt; and may contain up to 17.4% of their total weight in caffeine,&lt;ref&gt;{{Cite journal|last1=Seeram|first1=Navindra P.|last2=Henning|first2=Susanne M.|last3=Niu|first3=Yantao|last4=Lee|first4=Rupo|last5=Scheuller|first5=H. Samuel|last6=Heber|first6=David|date=2006-03-01|title=Catechin and Caffeine Content of Green Tea Dietary Supplements and Correlation with Antioxidant Capacity|journal=Journal of Agricultural and Food Chemistry|volume=54|issue=5|pages=1599–1603|doi=10.1021/jf052857r|pmid=16506807|issn=0021-8561}}&lt;/ref&gt; though decaffeinated versions are also available.&lt;ref&gt;{{cite web|title=Update on the USP Green Tea Extract Monograph|url=http://www.usp.org/usp-nf/notices/retired-compendial-notices/update-usp-green-tea-extract-monograph|publisher=USP|language=en|date=April 10, 2009}}&lt;/ref&gt;

Numerous claims have been made for the health benefits of green tea, but human clinical research has not found good evidence of benefit.&lt;ref name=&quot;nccih16&quot;&gt;{{cite web |title=Green tea |url=https://nccih.nih.gov/health/greentea |publisher=National Center for Complementary and Integrative Health, US National Institutes of Health |accessdate=12 August 2018 |date=September 2016 |quote=Green tea extracts haven't been shown to produce a meaningful weight loss in overweight or obese adults. They also haven't been shown to help people maintain a weight loss.}}&lt;/ref&gt;&lt;ref name=&quot;efsa&quot;&gt;{{cite web|url=http://www.efsa.europa.eu/en/efsajournal/pub/2055|publisher=European Food Safety Authority|title=Scientific Opinion on the substantiation of health claims related to Camellia sinensis (L.) Kuntze (tea), including catechins in green tea, and improvement of endothelium-dependent vasodilation (ID 1106, 1310), maintenance of normal blood pressure (ID 1310, 2657), maintenance of normal blood glucose concentrations (ID 1108), maintenance of normal blood LDL cholesterol concentrations (ID 2640), protection of the skin from UV-induced (including photo-oxidative) damage (ID 1110, 1119), protection of DNA from oxidative damage (ID 1120, 1121), protection of lipids from oxidative damage (ID 1275), contribution to normal cognitive function (ID 1117, 2812), &quot;cardiovascular system&quot; (ID 2814), &quot;invigoration of the body&quot; (ID 1274, 3280), decreasing potentially pathogenic gastro-intestinal microorganisms (ID 1118), &quot;immune health&quot; (ID 1273) and &quot;mouth&quot; (ID 2813) pursuant to Article 13(1) of Regulation (EC) No 1924/2006|date=8 April 2011|accessdate=9 November 2014}}&lt;/ref&gt;&lt;ref name=&quot;Filippini2020&quot;&gt;{{cite journal |last1=Filippini |first1=T |last2=Malavolti |first2=M |last3=Borrelli |first3=F |last4=Izzo |first4=AA |last5=Fairweather-Tait |first5=SJ |last6=Horneber |first6=M |last7=Vinceti |first7=M |title=Green tea (Camellia sinensis) for the prevention of cancer. |journal=Cochrane Database of Systematic Reviews |date=March 2020 |volume=3 |page=CD005004 |doi=10.1002/14651858.CD005004.pub3 |pmid=32118296 |pmc=7059963 }}&lt;/ref&gt; In 2011, a panel of scientists published a report on the claims for health effects at the request of the European Commission: in general they found that the claims made for green tea were not supported by sufficient scientific evidence.&lt;ref name=&quot;efsa&quot;/&gt; Although green tea may enhance mental alertness due to its caffeine content, there is only weak, inconclusive evidence that regular consumption of green tea affects the risk of cancer or cardiovascular diseases, and there is no evidence that it benefits weight loss.&lt;ref name=nccih16/&gt;

A 2020 review by the Cochrane Collaboration listed some potential adverse effects including gastrointestinal disorders, higher levels of liver enzymes, and, more rarely, insomnia, raised blood pressure, and skin reactions.&lt;ref name=coch2020/&gt;

Research has shown there is no good evidence that green tea helps to prevent or treat cancer in people.&lt;ref name=coch2020&gt;{{cite journal| author=Filippini T, Malavolti M, Borrelli F, Izzo AA, Fairweather-Tait SJ, Horneber M | display-author=etal| title=Green tea (Camellia sinensis) for the prevention of cancer. | journal=Cochrane Database Syst Rev | date= 2020 | volume= 3 | pages= CD005004 | pmid=32118296 | doi=10.1002/14651858.CD005004.pub3 |type=Systematic review | pmc=7059963}}&lt;/ref&gt;

The link between green tea consumption and the risk of certain cancers such as stomach cancer and non-melanoma skin cancers is unclear due to inconsistent or inadequate evidence.&lt;ref name=&quot;Hou2013&quot;&gt;{{cite journal|author=Hou IC, Amarnani S, Chong MT, Bishayee A |title=Green tea and the risk of gastric cancer: epidemiological evidence|journal=World J Gastroenterol|volume=19|issue=24|pages=3713–22|date=June 2013|pmid=23840110|pmc=3699047|doi=10.3748/wjg.v19.i24.3713|type=Review}}&lt;/ref&gt;&lt;ref name=&quot;Caini2017&quot;&gt;{{cite journal |last1=Caini |first1=S |last2=Cattaruzza |first2=MS |last3=Bendinelli |first3=B |last4=Tosti |first4=G |last5=Masala |first5=G |last6=Gnagnarella |first6=P |last7=Assedi |first7=M |last8=Stanganelli |first8=I |last9=Palli |first9=D |last10=Gandini |first10=S |title=Coffee, tea and caffeine intake and the risk of non-melanoma skin cancer: a review of the literature and meta-analysis |journal=European Journal of Nutrition |date=February 2017 |volume=56 |issue=1 |pages=1–12 |doi=10.1007/s00394-016-1253-6 |pmid=27388462|s2cid=24758243 |type=Systematic Review &amp; Meta-Analysis}}&lt;/ref&gt;

Green tea interferes with the chemotherapy drug bortezomib (Velcade) and other boronic acid-based proteasome inhibitors, and should be avoided by people taking these medications.&lt;ref&gt;{{cite journal|author=Jia L, Liu FT |title=Why bortezomib cannot go with 'green'?|journal=Cancer Biol Med|volume=10|issue=4|pages=206–13|date=December 2013|pmid=24349830|pmc=3860349|doi=10.7497/j.issn.2095-3941.2013.04.004|type=Review}}&lt;/ref&gt;

Observational studies found a minor correlation between daily consumption of green tea and a 5% lower risk of death from cardiovascular disease. In a 2015 meta-analysis of such observational studies, an increase in one cup of green tea per day was correlated with slightly lower risk of death from cardiovascular causes.&lt;ref name=&quot;Tang2015&quot;&gt;{{cite journal|author=Tang J, Zheng JS, Fang L, Jin Y, Cai W, Li D |title=Tea consumption and mortality of all cancers, CVD and all causes: a meta-analysis of eighteen prospective cohort studies|journal=Br J Nutr|volume=114|issue=5|pages=673–83|date=July 2015|pmid=26202661|doi=10.1017/S0007114515002329|type=Meta-analysis|doi-access=free}}&lt;/ref&gt; Green tea consumption may be correlated with a reduced risk of [[stroke]].&lt;ref name=&quot;Zhang2015&quot;&gt;{{cite journal|author=Zhang C, Qin YY, Wei X, Yu FF, Zhou YH, He J |title=Tea consumption and risk of cardiovascular outcomes and total mortality: a systematic review and meta-analysis of prospective observational studies|journal=Eur J Epidemiology|volume=30|issue=2|pages=103–13|date=February 2015|pmid=25354990|doi=10.1007/s10654-014-9960-x|s2cid=22529707|type=Systematic Review and Meta-Analysis}}&lt;/ref&gt;&lt;ref name=&quot;Kromhout2016&quot;&gt;{{cite journal |last1=Kromhout |first1=D |last2=Spaaij |first2=CJ |last3=de Goede |first3=J |last4=Weggemans |first4=RM |title=The 2015 Dutch food-based dietary guidelines |journal=European Journal of Clinical Nutrition |date=August 2016 |volume=70 |issue=8 |pages=869–78 |doi=10.1038/ejcn.2016.52 |pmid=27049034|pmc=5399142|type=Review}}&lt;/ref&gt; Meta-analyses of randomized controlled trials found that green tea consumption for 3–6 months may produce small reductions (about 2–3&amp;nbsp;mm Hg each) in systolic and diastolic blood pressures.&lt;ref name=&quot;Kromhout2016&quot;/&gt;&lt;ref name=&quot;Liu2014&quot;&gt;{{cite journal|author=Liu G, Mi XN, Zheng XX, Xu YL, Lu J, Huang XH |title=Effects of tea intake on blood pressure: a meta-analysis of randomised controlled trials|journal=Br J Nutr|volume=112|issue=7|pages=1043–54|date=October 2014|pmid=25137341|doi=10.1017/S0007114514001731|type=Meta-Analysis|doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;Khalesi2014&quot;&gt;{{cite journal|author=Khalesi S, Sun J, Buys N, Jamshidi A, Nikbakht-Nasrabadi E, Khosravi-Boroujeni H |title=Green tea catechins and blood pressure: a systematic review and meta-analysis of randomised controlled trials|journal=Eur J Nutr|volume=53|issue=6|pages=1299–1311|date=September 2014|pmid=24861099|doi=10.1007/s00394-014-0720-1|s2cid=206969226|type=Systematic Review and Meta-Analysis}}&lt;/ref&gt;&lt;ref name=&quot;Mozaffarian2016&quot;&gt;{{cite journal |last1=Mozaffarian |first1=D |title=Dietary and Policy Priorities for Cardiovascular Disease, Diabetes, and Obesity: A Comprehensive Review |journal=Circulation |date=January 2016 |volume=133 |issue=2 |pages=187–225 |doi=10.1161/CIRCULATIONAHA.115.018585 |pmid=26746178|pmc=4814348|type=Review}}&lt;/ref&gt; A separate systematic review and meta-analysis of randomized controlled trials found that consumption of 5-6 cups of green tea per day was associated with a small reduction in systolic blood pressure (2 mmHg), but did not lead to a significant difference in diastolic blood pressure.&lt;ref name=&quot;Onakpoya2014&quot;&gt;{{cite journal |last1=Onakpoya |first1=I |last2=Spencer |first2=E |last3=Heneghan |first3=C |last4=Thompson |first4=M |title=The effect of green tea on blood pressure and lipid profile: a systematic review and meta-analysis of randomized clinical trials |journal=Nutrition, Metabolism and Cardiovascular Diseases |date=August 2014 |volume=24 |issue=8 |pages=823–36 |doi=10.1016/j.numecd.2014.01.016 |pmid=24675010|type=Systematic Review &amp; Meta-Analysis}}&lt;/ref&gt;

Green tea consumption lowers fasting glucose (fasting blood sugar) but in clinical studies the beverage's effect on hemoglobin A1c and fasting insulin levels was inconsistent.&lt;ref name=&quot;Larsson2014&quot;&gt;{{cite journal|author=Larsson SC|title=Coffee, tea, and cocoa and risk of stroke|journal=Stroke|volume=45|issue=1|pages=309–14|date=January 2014|pmid=24326448|doi=10.1161/STROKEAHA.113.003131|type=Review|doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;Glycemic2013&quot;&gt;{{cite journal|author=Liu K, Zhou R, Wang B, Chen K, Shi LY, Zhu JD, Mi MT |title=Effect of green tea on glucose control and insulin sensitivity: a meta-analysis of 17 randomized controlled trials|journal=Am J Clin Nutr|volume=98|issue=2|pages=340–8|date=August 2013|pmid=23803878|doi=10.3945/ajcn.112.052746|type=Meta-Analysis|doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;ZhengGlycemic2013&quot;&gt;{{cite journal|author=Zheng XX, Xu YL, Li SH, Hui R, Wu YJ, Huang XH |title=Effects of green tea catechins with or without caffeine on glycemic control in adults: a meta-analysis of randomized controlled trials|journal=Am J Clin Nutr|volume=97|issue=4|pages=750–62|date=April 2013|pmid=23426037|doi=10.3945/ajcn.111.032573|type=Meta-Analysis|doi-access=free}}&lt;/ref&gt;

Drinking green tea or taking green tea supplements decreases the blood concentration of total cholesterol (about 3–7&amp;nbsp;mg/dL), low density lipoprotein (LDL cholesterol) (about 2&amp;nbsp;mg/dL), and does not affect the concentration of high density lipoprotein (HDL cholesterol) or triglycerides.&lt;ref name=&quot;Onakpoya2014&quot;/&gt;&lt;ref name=&quot;Larsson2014&quot;/&gt;&lt;ref name=&quot;ZhengCholesterol2011&quot;&gt;{{cite journal|author=Zheng XX, Xu YL, Li SH, Liu XX, Hui R, Huang XH |title=Green tea intake lowers fasting serum total and LDL cholesterol in adults: a meta-analysis of 14 randomized controlled trials|journal=Am J Clin Nutr|volume=94|issue=2|pages=601–10|date=August 2011|pmid=21715508|doi=10.3945/ajcn.110.010926|type=Meta-Analysis|doi-access=free}}&lt;/ref&gt; A 2013 Cochrane meta-analysis of longer-term randomized controlled trials (&gt;3 months duration) concluded that green tea consumption lowers total and LDL cholesterol concentrations in the blood.&lt;ref name=&quot;Larsson2014&quot;/&gt;

A 2015 systematic review and meta-analysis of 11 randomized controlled trials found that green tea consumption was not significantly associated with lower plasma levels of C-reactive protein levels (a marker of inflammation).&lt;ref name=&quot;Serban2015&quot;&gt;{{cite journal|author=Serban C, Sahebkar A, Antal D, Ursoniu S, Banach M |title=Effects of supplementation with green tea catechins on plasma C-reactive protein concentrations: A systematic review and meta-analysis of randomized controlled trials|journal=Nutrition|volume=31|issue=9|pages=1061–71|date=September 2015|pmid=26233863|doi=10.1016/j.nut.2015.02.004|type=Systematic review &amp; meta-analysis}}&lt;/ref&gt;

There is no good evidence that green tea aids in weight loss or weight maintenance.&lt;ref name=nccih16/&gt;&lt;ref name=weight&gt;{{cite journal |author=Jurgens TM, Whelan AM, Killian L, Doucette S, Kirk S, Foy E |title=Green tea for weight loss and weight maintenance in overweight or obese adults |journal=Cochrane Database Syst Rev |volume=12 |pages=CD008650 |date=2012 |pmid=23235664 |doi=10.1002/14651858.CD008650.pub2 |type=Systematic review}}&lt;/ref&gt;

Excessive consumption of green tea extract has been associated with hepatotoxicity and liver failure.&lt;ref&gt;{{cite web | url = https://livertox.nih.gov/GreenTea.htm | title = Green Tea | work = LiverTox: Clinical and Research Information on Drug-Induced Liver Injury | publisher = National Institutes of Health | quote = Green tea extract and, more rarely, ingestion of large amounts of green tea have been implicated in cases of clinically apparent acute liver injury, including instances of acute liver failure and either need for urgent liver transplantation or death.}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal | doi = 10.1007/s00204-015-1521-x| pmid = 25975988| title = Hepatotoxicity of green tea: An update| journal = Archives of Toxicology| volume = 89| issue = 8| pages = 1175–1191| date = 2015| last1 = Mazzanti| first1 = Gabriela| last2 = Di Sotto| first2 = Antonella| last3 = Vitalone| first3 = Annabella| s2cid = 14744653}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author=Javaid A, Bonkovsky HL| title = Hepatotoxicity due to extracts of Chinese green tea (''Camellia sinensis''): a growing concern | journal = J Hepatol | date = 2006 | volume = 45 | issue = 2 | pages = 334–336| doi = 10.1016/j.jhep.2006.05.005 | pmid = 16793166 }}&lt;/ref&gt; In 2018, a scientific panel for the European Food Safety Authority reviewed the safety of green tea consumption over a low-moderate range of daily EGCG intake from 90 to 300&amp;nbsp;mg per day, and with exposure from high green tea consumption estimated to supply up to 866&amp;nbsp;mg EGCG per day.&lt;ref name=&quot;efsa18 &quot;&gt;{{cite journal |author=EFSA Panel on Food Additives and Nutrient Sources added to Food | title=Scientific opinion on the safety of green tea catechins | journal=EFSA Journal | volume=16 | issue=4 | date=2018 | pages=e05239 | issn=1831-4732 | doi=10.2903/j.efsa.2018.5239 | pmid=32625874 | pmc=7009618 | doi-access=free }}&lt;/ref&gt; Dietary supplements containing EGCG may supply up to 1000&amp;nbsp;mg EGCG and other catechins per day.&lt;ref name=efsa18/&gt; The panel concluded that EGCG and other catechins from green tea in low-moderate daily amounts are generally regarded as safe, but in some cases of excessive consumption of green tea or use of high-EGCG supplements, liver toxicity may occur.&lt;ref name=efsa18/&gt;

==''Cassia abbreviata''==
{{main|Remedy/Plants/Fabaceae}}

==''Cassia javanica''==
{{main|Remedy/Plants/Fabaceae}}

==''Chamaemelum nobile''==
{{main|Remedy/Plants/Asteraceae}}

==''Chondrus crispus''==
[[Image:Life cycle Chondrus.jpg|thumb|right|250px|The lifecycle of ''Chondrus crispus'': Below the life stages are indicated if the life stage is haploid (n) or diploid (2n) and the type of carrageenan present. Credit: [[c:user:Chondrus|Chondrus]].{{tlx|free media}}]]
[[Image:Chondrus crispus.jpg|thumb|left|250px|How the lifecycles of ''C. crispus'' might look in nature: The gametophytes show blue iridescence and the fertile sporophytes exhibit a spotty pattern. Credit: [[c:user:Kontos|Kontos]].{{tlx|free media}}]]
Family: ''Gigartinaceae''.

''C. crispus'' is an industrial source of carrageenan commonly used as a thickener and stabilizer in milk products, such as ice cream and processed foods.&lt;ref name=Carrageenan&gt;{{ cite web |title=Carrageenan |url=https://pubchem.ncbi.nlm.nih.gov/compound/Carrageenan |publisher=PubChem, US National Library of Medicine |accessdate=11 January 2021 |date=9 January 2021 }}&lt;/ref&gt;

Carrageenan, E407 or E407a, may be used as a thickener in calico printing and paper marbling, and for fining beer.&lt;ref name=Carrageenan/&gt;&lt;ref&gt;Chisholm, Hugh, ed. (1911). &quot;Irish Moss&quot; . Encyclopædia Britannica. Vol. 14 (11th ed.). Cambridge University Press. p. 795.&lt;/ref&gt; Irish moss is frequently used with ''Mastocarpus stellatus'' (''Gigartina mamillosa''), ''Chondracanthus acicularis'' (''G. acicularis''), and other seaweeds, which are all commonly found growing together. Carrageenan may be extracted from tropical seaweeds of the genera ''Kappaphycus'' and ''Eucheuma''.&lt;ref name=Bixler&gt;{{cite journal | last1 = Bixler | first1 = H. J. | last2 = Porse | first2 = H. | year = 2011 | title = A decade of change in the seaweed hydrocolloids industry | journal = Journal of Applied Phycology | volume = 23 | issue = 3| pages = 321–335 | doi=10.1007/s10811-010-9529-3| s2cid = 24607698 }}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot; |+ Seaweed, irishmoss, raw
|-
! Nutrition !! Nutritional value per 100 g (3.5 oz) 
|-
| kJ || 205 
|-
| protein || 1.51 g 
|-
| fat || 0.16 g 
|-
| carbs || 12.29 g 
|-
| fiber || 1.3 g 
|-
| sugars || 0.61 g
|-
| calcium || 72 mg 
|-
| iron || 8.9 mg 
|-
| magnesium || 144 mg 
|-
| phosphorus || 157 mg 
|-
| sodium || 61 mg 
|-
| zinc || 1.95 mg 
|-
| manganese || 0.37 mg 
|-
| vit C || 3 mg 
|-
| riboflavin || 0.466 mg 
|-
| niacin || 0.593 mg 
|-
| pantothenic || 0.176 mg 
|-
| vitB6 || 0.069 mg 
|-
| folate || 182 µg 
|-
| vitE || 0.87 mg 
|-
| vitK || 5 µg 
|-
| source [http://ndb.nal.usda.gov/ndb/search/list?qlookup=11444&amp;format=Full Link to USDA Database entry]
|}

The nuclear genome&lt;ref name=Collen&gt;{{ cite journal | last1 = Collén | first1 = J | year = 2013 | title = Genome structure and metabolic features in the red seaweed ''Chondrus crispus'' shed light on evolution of the ''Archaeplastida'' | journal = Proceedings of the National Academy of Sciences | volume =  110| issue = 13| pages =  5247–5252| doi = 10.1073/pnas.1221259110 | pmid=23503846 | pmc=3612618| bibcode = 2013PNAS..110.5247C }}&lt;/ref&gt; is 105 Mbp and is coding for 9,606 genes, characterised by relatively few genes with very few introns, where genes are clustered together, with normally short distances between genes and then large distances between groups of genes.

Genetic &quot;deletion of fatty acid amide hydrolase (FAAH), the enzyme responsible for degradation of fatty acid amides, including [...] N-palmitoyl ethanolamine (PEA), N-oleoyl ethanolamine (OEA), and oleamide, also elicits anti-edema&quot;.&lt;ref name=Wise&gt;{{ cite journal
|author=Laura E. Wise, Roberta Cannavacciulo, Benjamin F. Cravatt, Billy F. Martin, and Aron H. Lichtman
|title=Evaluation of fatty acid amides in the carrageenan-induced paw edema model
|journal=Neuropharmacology
|date=January 2008
|volume=54
|issue=1
|pages=181-188
|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2200792/
|arxiv=
|bibcode=
|doi=10.1016/j.neuropharm.2007.06.003
|pmid=17675189
|accessdate=10 February 2022 }}&lt;/ref&gt;

&quot;PEA, [dexamethasone] DEX, [diclofenac] DIC elicited significant decreases in carrageenan-induced paw edema in wild type mice.&quot;&lt;ref name=Wise/&gt;

&quot;OEA produced a less reliable anti-edema effect than these other drugs, and [N-arachidonoyl ethanolamine, anandamide] AEA and oleamide failed to produce any significant decreases in paw edema.&quot;&lt;ref name=Wise/&gt;

&quot;PEA administration and FAAH blockade elicited anti-edema effects of an equivalent magnitude&quot;.&lt;ref name=Wise/&gt;
{{clear}}

==''Cichorium intybus''==
{{main|Remedy/Plants/Asteraceae}}

==''Cinnamomum burmannii''==

Family ''Lauraceae''.

''Cinnamomum burmannii'' is Korintje, Padang cassia, or Indonesian cinnamon.

==''Cinnamomum cassia''==

Cassia or Chinese cinnamon is the most common commercial type in the USA.

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot;
|+ Cinnamon, spice, ground 
|-
! Energy !! 1,035 kJ (247 kcal) 
|-
! !! Nutritional value per 100 g (3.5 oz) 
|-
! Carbohydrates !! 80.6 
|-
| Sugars || 2.2 
|-
| Dietary fiber || 53.1 
|-
! Fat !! 1.2 
|-
! Protein !! 4 
|-
! Vitamins !! Quantity !! % Daily value (DV)* 
|-
| Vitamin A equivalent || 15 µg || 2 
|-
| Thiamine (B&lt;sub&gt;1&lt;/sub&gt;) || 0.02 mg || 2 
|-
| Riboflavin (B&lt;sub&gt;2&lt;/sub&gt;) || 0.04 mg || 3 
|-
| Niacin (B&lt;sub&gt;3&lt;/sub&gt;) || 1.33 mg || 9 
|-
| Pyridoxine B&lt;sub&gt;6&lt;/sub&gt; || 0.16 mg || 12 
|-
| Folate B&lt;sub&gt;9&lt;/sub&gt; || 6 µg || 2 
|-
| Vitamin C || 3.8 mg || 5 
|-
| Vitamin E || 2.3 mg || 15 
|-
| Vitamin K || 31.2 µg || 30 
|-
! Minerals !! Quantity !! % DV* 
|-
| Calcium || 1002 mg || 100 
|-
| Iron || 8.3 mg || 64 
|-
| Magnesium || 60 mg || 17 
|-
| Phosphorus || 64 mg || 9 
|-
| Potassium || 431 mg || 9 
|-
| Sodium || 10 mg || 1 
|-
| Zinc || 1.8 mg || 19 
|-
! Other constituents !! Quantity 
|-
| Water || 10.6 gm 
|-
|}
* note: Source: USDA Database&lt;ref name=USDAdata&gt;{{cite web|url=https://ndb.nal.usda.gov/fdc-app.html#/food-details/171320/nutrients|date=1 April 2019|title=Spices, cinnamon, ground|website=FoodData Central|publisher=Agricultural Research Service|url-status=live|archive-url=https://archive.today/20200905232856/https://ndb.nal.usda.gov/fdc-app.html%23/food-details/171320/nutrients#/food-details/171320/nutrients|archive-date=5 September 2020|accessdate=6 September 2020}}&lt;/ref&gt;

Ground cinnamon is composed of around 11% water, 81% carbohydrates (including 53% dietary fiber), 4% protein, and 1% fat.&lt;ref name=USDAdata/&gt; In a 100 gram reference amount, ground cinnamon is a rich source of calcium (100% of the Daily Value (DV)), iron (64% DV), and vitamin K (30% DV).

==''Cinnamomum citriodorum''==

''Cinnamomum citriodorum'' is Malabar cinnamon.

==''Cinnamomum loureiroi''==

''Cinnamomum loureiroi'' is Saigon cinnamon, Vietnamese cassia, or Vietnamese cinnamon.

==''Cinnamomum verum''==
[[Image:Cinnamomum verum spices.jpg|thumb|right|300px|Dried bark strips, bark powder and flowers of the small tree ''Cinnamomum verum'' are shown. Credit: [[c:user:LivingShadow|Simon A. Eugster]].{{tlx|free media}}]]
''Cinnamomum verum'' is Sri Lanka cinnamon, Ceylon cinnamon or ''Cinnamomum zeylanicum''.

Cinnamon is a spice obtained from the inner bark of several tree species from the genus ''Cinnamomum'', used mainly as an aromatic condiment and flavouring additive in a wide variety of cuisines, sweet and savoury dishes, breakfast cereals, snackfoods, tea and traditional foods, derived from its essential oil and principal component, cinnamaldehyde, as well as numerous other constituents including eugenol.

&quot;Among the identified compounds, trans-cinnamaldehyde and p-cymene significantly reduced the [lipopolysaccharides] LPS-dependent IL-8 secretion in THP-1 monocytes. Synergistic anti-inflammatory effects were observed for combinations of trans-cinnamaldehyde with p-cymene, cinnamyl alcohol or cinnamic acid. Moreover, cinnamon extract as well as trans-cinnamaldehyde and p-cymene mitigated the phosphorylation of [Protein kinase B] Akt and [NF-κB inhibitor alpha] IκBα. [...] Trans-cinnamaldehyde and p-cymene contribute to the strong anti-inflammatory effects of cinnamon extract. [S]ynergistic effects among compounds that do not exhibit anti-inflammatory effects themselves might be present to positively influence the beneficial effects of cinnamon bark extract.&quot;&lt;ref name=Schink&gt;{{ cite journal
|author=Anne Schink, Katerina Naumoska, Zoran Kitanovski, Christopher Johannes Kampf, Janine Fröhlich-Nowoisky, Eckhard Thines, Ulrich Pöschl, Detlef Schuppan and Kurt Lucas
|title=Anti-inflammatory effects of cinnamon extract and identification of active compounds influencing the TLR2 and TLR4 signaling pathways
|journal=Food &amp; Function
|date=25 October 2018
|volume=9
|issue=
|pages=5950-5964
|url=https://pubs.rsc.org/en/content/articlehtml/2018/fo/c8fo01286e
|arxiv=
|bibcode=
|doi=10.1039/C8FO01286E
|pmid=
|accessdate=10 February 2022 }}&lt;/ref&gt;
{{clear}}

==''Citrus maxima''==

The '''orange''' is the fruit of various ''Citrus'' species in the family ''Rutaceae''; it primarily refers to ''Citrus'' × ''sinensis'',&lt;ref name=&quot;USDACitrus&quot;&gt;{{cite web
|title=Citrus ×sinensis (L.) Osbeck (pro sp.) (maxima × reticulata) sweet orange |work=Plants.USDA.gov
|url=https://web.archive.org/web/20110512023634/http://plants.usda.gov/java/profile?symbol=CISI3 |archive-date=May 12, 2011 }}&lt;/ref&gt; which is also called sweet orange, to distinguish it from the related ''Citrus × aurantium'', referred to as bitter orange. The sweet orange reproduces asexually (apomixis through nucellar embryony); varieties of sweet orange arise through mutations.&lt;ref name=fullgenome /&gt;&lt;ref&gt;{{cite web|url=https://wikifarmer.com/orange-fruit-information/|title=Orange Fruit Information|date=9 June 2017|accessdate=20 September 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=https://www.nutrition-and-you.com/orange-fruit.html|title=Orange fruit nutrition facts and health benefits|access-date=20 September 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=https://www.livescience.com/45057-oranges-nutrition-facts.html|title=Oranges: Health Benefits, Risks &amp; Nutrition Facts|access-date=20 September 2018}}&lt;/ref&gt;

The orange is a hybrid between pomelo (''Citrus maxima'') and mandarin (''Citrus reticulata'').&lt;ref name=fullgenome&gt;{{cite journal | doi = 10.1038/ng.2472 | volume=45 | title=The draft genome of sweet orange (Citrus sinensis) | journal=Nature Genetics | pages=59–66 | pmid=23179022 | date=Jan 2013 | last1 = Xu | first1 = Q | last2 = Chen | first2 = LL | last3 = Ruan | first3 = X | last4 = Chen | first4 = D | last5 = Zhu | first5 = A | last6 = Chen | first6 = C | last7 = Bertrand | first7 = D | last8 = Jiao | first8 = WB | last9 = Hao | first9 = BH | last10 = Lyon | first10 = MP | last11 = Chen | first11 = J | last12 = Gao | first12 = S | last13 = Xing | first13 = F | last14 = Lan | first14 = H | last15 = Chang | first15 = JW | last16 = Ge | first16 = X | last17 = Lei | first17 = Y | last18 = Hu | first18 = Q | last19 = Miao | first19 = Y | last20 = Wang | first20 = L | last21 = Xiao | first21 = S | last22 = Biswas | first22 = MK | last23 = Zeng | first23 = W | last24 = Guo | first24 = F | last25 = Cao | first25 = H | last26 = Yang | first26 = X | last27 = Xu | first27 = XW | last28 = Cheng | first28 = YJ | last29 = Xu | first29 = J | last30 = Liu | first30 = JH | last31 = Luo | first31 = OJ | last32 = Tang | first32 = Z | last33 = Guo | first33 = WW | last34 = Kuang | first34 = H | last35 = Zhang | first35 = HY | last36 = Roose | first36 = ML | last37 = Nagarajan | first37 = N | last38 = Deng | first38 = XX | last39 = Ruan | first39 = Y| issue=1 }}&lt;/ref&gt;&lt;ref name=&quot;AGLthesis&quot;&gt;{{cite thesis |title=Organización de la diversidad genética de los cítricos |year=2013|author=Andrés García Lor| url=https://riunet.upv.es/bitstream/handle/10251/31518/Versión3.Tesis%20Andrés%20García-Lor.pdf |pages=79 }}&lt;/ref&gt; The chloroplast genome, and therefore the maternal line, is that of pomelo.&lt;ref name=genealogy_review&gt;{{cite journal | doi = 10.1038/nbt.2954 | volume=32 | issue=7 | title=A genealogy of the citrus family | journal=Nature Biotechnology | pages=640–642 | pmid=25004231 | last1 = Velasco | first1 = R | last2 = Licciardello | first2 = C| year=2014 | s2cid=9357494 | doi-access=free }}&lt;/ref&gt; The sweet orange has had its full genome sequenced.&lt;ref name=fullgenome /&gt;

Oranges contain diverse phytochemicals, including carotenoids (beta-carotene, lutein and beta-cryptoxanthin), flavonoids (e.g. naringenin)&lt;ref&gt;{{cite journal|journal=J Agric Food Chem|year=2015|volume=63|issue=2|title=In Vitro Bioaccessibility of Carotenoids, Flavonoids, and Vitamin C from Differently Processed Oranges and Orange Juices [Citrus sinensis (L.) Osbeck]|authors=Aschoff JK, Kaufmann S, Kalkan O, Neidhart S, Carle R, Schweiggert RM|pmid=25539394|doi=10.1021/jf505297t|pages=578–87}}&lt;/ref&gt; and numerous volatile organic compounds producing orange aroma, including aldehydes, esters, terpenes, alcohols, and ketones.&lt;ref&gt;{{cite journal|journal=Crit Rev Food Sci Nutr|year=2008|volume=48|issue=7|pages=681–95|doi=10.1080/10408390701638902|title=Fresh squeezed orange juice odor: a review|authors=Perez-Cacho PR, Rouseff RL|pmid=18663618|s2cid=32567584}}&lt;/ref&gt;

Although not as juicy or tasty as the flesh, orange peel is edible and has significant contents of vitamin C, dietary fiber, total polyphenols, carotenoids, limonene and dietary minerals, such as potassium and magnesium.&lt;ref&gt;{{cite journal|journal=Food Chem|year=2012|volume=134|issue=4|pages=1892–8|doi=10.1016/j.foodchem.2012.03.090|title=Antioxidant capacity and mineral content of pulp and peel from commercial cultivars of citrus from Brazil|authors=Barros HR, Ferreira TA, Genovese MI|pmid=23442635}}&lt;/ref&gt;

==''Citrus reticulata''==

==''Coffea arabica''==
{{main|Remedy/Plants/Rubiaceae}}

==''Coffea canephora''==
{{main|Remedy/Plants/Rubiaceae}}

==''Coffea liberica''==
{{main|Remedy/Plants/Rubiaceae}}

==''Coffea racemosa''==
{{main|Remedy/Plants/Rubiaceae}}

==''Commiphora myrrha''==
[[Image:Commiphora-myrrha-resin-myrrh.jpg|thumb|right|Small lumps are myrrh resin. Credit: [[w:user:Sjschen|Sjschen]].{{tlx|free media}}]]
Family ''Burseraceae''.

Myrrh is a natural gum-resin extracted from a number of small, thorny tree species of the genus ''Commiphora''.&lt;ref&gt;Rice, Patty C., ''Amber: Golden Gem of the Ages'', Author House, Bloomington, 2006 p.321&lt;/ref&gt; Myrrh resin has been used throughout history as a perfume, incense and medicine. Myrrh mixed with posca or wine was common across ancient cultures, for general pleasure, and as an analgesic.&lt;ref&gt;Pliny the Elder [-79 CE], trans. John Bostock and Henry Thomas Riley, &quot;Wines Drunk by the Ancient Romans&quot;, ''The Natural History'' [c. 77 CE], book 14, ch. 15. London: H.G. Bohn, 1855. 253.,Available online at books.google.com/books?id=A0EMAAAAIAAJ&amp;pg=PA253&lt;/ref&gt;

Myrrh is used as an antiseptic in mouthwashes, gargles, and toothpastes.&lt;ref&gt;{{cite web|url=http://www.worldagroforestrycentre.org/sea/products/afdbases/af/asp/SpeciesInfo.asp?SpID=17990 |archive-url=https://web.archive.org/web/20110930043102/http://www.worldagroforestrycentre.org/sea/products/afdbases/af/asp/SpeciesInfo.asp?SpID=17990 |url-status=dead |archive-date=2011-09-30 |title=Species Information |publisher=www.worldagroforestrycentre.org |accessdate=2009-01-15 }}&lt;/ref&gt; It is also used in some liniments and healing salves that may be applied to abrasions and other minor skin ailments. Myrrh has been used as an analgesic for toothaches and can be used in liniment for bruises, aches, and sprains.&lt;ref name=&quot;trieste&quot;&gt;{{cite web|url=http://www.ics.trieste.it/MAPs/MedicinalPlants_Plant.aspx?id=599|archive-url=https://archive.today/20110809204252/http://www.ics.trieste.it/MAPs/MedicinalPlants_Plant.aspx?id=599|url-status=dead|archive-date=2011-08-09|title=ICS-UNIDO – MAPs|publisher=www.ics.trieste.it|accessdate=2009-01-16}}&lt;/ref&gt;

Myrrh gum is commonly claimed to remedy indigestion, ulcers, colds, cough, asthma, lung congestion, arthritis pain, and cancer.&lt;ref name=&quot;faraj&quot;&gt;{{Cite journal|pmid=15814041|date=2005|last1=Al Faraj|first1=S|title=Antagonism of the anticoagulant effect of warfarin caused by the use of Commiphora molmol as a herbal medication: A case report|volume=99|issue=2|pages=219–20|doi=10.1179/136485905X17434|journal=Annals of Tropical Medicine and Parasitology|s2cid=2097777}}&lt;/ref&gt;

Myrrh is said to have special efficacy on the heart, liver, and spleen meridians as well as &quot;blood-moving&quot; powers to purge stagnant blood from the uterus, recommended for rheumatic, arthritic, and circulatory problems, and for amenorrhea, dysmenorrhea, menopause and uterine tumours, uses similar to those of frankincense, with which it is often combined in decoctions, liniments, and incense, used in concert, myrrh is &quot;blood-moving&quot; while frankincense is thought to move the ''qi'', making it used for arthritic conditions, or combined with such herbs as notoginseng, safflower petals, angelica sinensis, cinnamon, and salvia miltiorrhiza, usually in alcohol, and used both internally and externally.&lt;ref&gt;{{Cite web|url=https://planetherbs.com/research-center/specific-herbs-articles/the-emmenagogues/#MYRRH|title=The Emmenagogues: Herbs that move blood and relieve pain: Myrrh|last=Tierra|first=Michael|date=June 3, 2019|website=East West School of Planetary Herbology|language=en-US|accessdate=2019-06-05}}&lt;/ref&gt;
{{clear}}

==''Commiphora wightii''==

Guggul (''Commiphora wightii'') is considered one of the best substances for the treatment of circulatory problems, nervous system disorders, and rheumatic complaints.&lt;ref&gt;[http://www.swsbm.com/ManualsMM/HRBENRGT.pdf Michael Moore ''Materia Medica'']&lt;/ref&gt;&lt;ref&gt;Tillotson, A., Chrysalis Natural Medicine Clinic, Myrrh Gum (Commiphora myrrha)] https://web.archive.org/web/20070614230838/http://oneearthherbs.squarespace.com/important-herbs/myrrh-gum-commiphora-myrrha.html 2007-06-14&lt;/ref&gt;

==''Coptis chinensis''==

Family ''Ranunculaceae''.

Berberine is found in ''Coptis chinensis'' (Chinese goldthread).

The rhizomes of ''Coptis chinensis'' contain the isoquinoline alkaloids berberine,&lt;ref&gt;[https://phytochem.nal.usda.gov/phytochem/plants/show/522 Dr. Duke's Phytochemical and Ethnobotanical Databases]&lt;/ref&gt; palmatine, and coptisine among others.

==''Corylus avellana''==
[[Image:Corylus avellana.jpg|thumb|right|250px|Common Hazel leaves and nuts are shown for ''Corylus avellana''. Credit: [[w:user:MPF|MPF]].{{tlx|free media}}]]
Family:	''Betulaceae''.

Hazelnuts are rich in protein and unsaturated fat, contain significant amounts of manganese, copper, vitamin E, thiamine, and magnesium.&lt;ref&gt;''SELF'' Nutrition data, [http://nutritiondata.self.com/facts/nut-and-seed-products/3116/2 Nuts, hazelnuts or filberts]. Accessed 2014-08-22.&lt;/ref&gt;
{{clear}}

==''Curcuma longa''==
[[Image:curcuminKeto.svg|right|frame|200px|Curcumin keto form is diagrammed. Credit: [[c:User:Ronhjones|Ronhjones]].{{tlx|free media}}]]
[[Image:curcumin.svg|right|frame|200px|Curcumin enol form is diagrammed. Credit: [[c:User:Ronhjones|Ronhjones]].{{tlx|free media}}]]
Family ''Zingiberaceae''.

Turmeric powder is about 60&amp;ndash;70% carbohydrates, 6&amp;ndash;13% water, 6&amp;ndash;8% protein, 5&amp;ndash;10% fat, 3&amp;ndash;7% dietary minerals, 3&amp;ndash;7% essential oils, 2&amp;ndash;7% dietary fiber, and 1&amp;ndash;6% curcuminoids.&lt;ref name=&quot;nelson&quot;&gt;{{cite journal |last1=Nelson |first1=KM |last2=Dahlin |first2=JL |last3=Bisson |first3=J |last4=Graham |first4=J |last5=Pauli |first5=GF |last6=Walters |first6=MA |display-author=3 |date=2017 |title=The Essential Medicinal Chemistry of Curcumin: Miniperspective |journal=Journal of Medicinal Chemistry |volume=60 |issue=5 |pmc=5346970 |pages=1620–1637 |doi=10.1021/acs.jmedchem.6b00975 |pmid=28074653 |quote=None of these studies [has] yet led to the approval of curcumin, curcuminoids, or turmeric as a therapeutic for any disease}}&lt;/ref&gt;

Phytochemical components of turmeric include diarylheptanoids, a class including numerous curcuminoids, such as curcumin, demethoxycurcumin, and bisdemethoxycurcumin.&lt;ref name=nelson/&gt;&lt;ref name=&quot;pubchem2020&quot;&gt;{{cite web |title=Curcumin |url=https://pubchem.ncbi.nlm.nih.gov/compound/969516 |publisher=PubChem, US National Library of Medicine |accessdate=25 November 2020 |date=21 November 2020}}&lt;/ref&gt; Curcumin constitutes up to 3.14% of assayed commercial samples of turmeric powder (the average was 1.51%); curry powder contains much less (an average of 0.29%).&lt;ref&gt;{{cite journal |author=Tayyem RF, Heath DD, Al-Delaimy WK, Rock CL |title= Curcumin content of turmeric and curry powders |journal= Nutr Cancer |volume= 55|issue=2 | pages= 126–131 |date= 2006 |pmid=17044766 |doi= 10.1207/s15327914nc5502_2|s2cid= 12581076 }}&lt;/ref&gt; Some 34 essential oils are present in turmeric, among which turmerone, germacrone, atlantone, and zingiberene are major constituents.&lt;ref&gt;{{cite journal|pmc=4142718|date=2014|last1=Hong|first1=SL|title=Essential Oil Content of the Rhizome of ''Curcuma purpurascens'' Bl. (Temu Tis) and Its Antiproliferative Effect on Selected Human Carcinoma Cell Lines|journal=The Scientific World Journal|volume=2014|pages=1–7|last2=Lee|first2=G. S|last3=Syed Abdul Rahman|first3=SN|last4=Ahmed Hamdi|first4=OA|last5=Awang|first5=K|last6=Aznam Nugroho|first6=N|last7=Abd Malek|first7=SN|doi=10.1155/2014/397430|pmid=25177723|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|pmid=24311554|date=2014|last1=Hu|first1=Y|title=GC-MS combined with chemometric techniques for the quality control and original discrimination of ''Curcumae longae'' rhizome: Analysis of essential oils|journal=Journal of Separation Science|volume=37|issue=4|pages=404–11|last2=Kong|first2=W|last3=Yang|first3=X|last4=Xie|first4=L|last5=Wen|first5=J|last6=Yang|first6=M|doi=10.1002/jssc.201301102}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|pmid=14558784|date=2003|last1=Braga|first1=ME|title=Comparison of yield, composition, and antioxidant activity of turmeric (''Curcuma longa'' L.) extracts obtained using various techniques|journal=Journal of Agricultural and Food Chemistry|volume=51|issue=22|pages=6604–11|last2=Leal|first2=PF|last3=Carvalho|first3=JE|last4=Meireles|first4=MA |doi=10.1021/jf0345550}}&lt;/ref&gt;

Turmeric and curcumin have been studied in numerous clinical trials for various human diseases and conditions, with no high-quality evidence of any anti-disease effect or health benefit.&lt;ref name=nelson/&gt;&lt;ref name=nccih2020/&gt;&lt;ref&gt;{{cite journal|pmid=27533649|pmc=5003001|date=2016|last1=Daily|first1=JW|title=Efficacy of Turmeric Extracts and Curcumin for Alleviating the Symptoms of Joint Arthritis: A Systematic Review and Meta-Analysis of Randomized Clinical Trials|journal=Journal of Medicinal Food|volume=19|issue=8|pages=717–29|last2=Yang|first2=M|last3=Park|first3=S|doi=10.1089/jmf.2016.3705}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|pmid=27213821|date=2016|last1=Vaughn|first1=A. R.|title=Effects of Turmeric (Curcuma longa) on Skin Health: A Systematic Review of the Clinical Evidence|journal=Phytotherapy Research|volume=30|issue=8|pages=1243–64|last2=Branum|first2=A|last3=Sivamani|first3=RK
|doi=10.1002/ptr.5640|s2cid=46429012}}&lt;/ref&gt; There is no scientific evidence that curcumin reduces inflammation.&lt;ref name=nelson/&gt;&lt;ref name=&quot;nccih2020&quot;&gt;{{cite web|title=Turmeric|url=https://www.nccih.nih.gov/health/turmeric|publisher=National Center for Complementary and Integrative Health, US National Institutes of Health|accessdate=25 November 2020|date=May 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author=White CM, Pasupuleti V, Roman YM, Li Y, Hernandez AV |title=Oral turmeric/curcumin effects on inflammatory markers in chronic inflammatory diseases: A systematic review and meta-analysis of randomized controlled trials |journal=Pharmacol Res |volume=146 |pages=104280 |date=August 2019 |pmid=31121255 |doi=10.1016/j.phrs.2019.104280 |type=Meta-analysis}}&lt;/ref&gt; There is weak evidence that turmeric extracts may be beneficial for relieving symptoms of knee osteoarthritis.&lt;ref&gt;{{cite journal |author=Wang Z, Singh A, Jones G, Winzenberg T, Ding C, Chopra A, Das S, Danda D, Laslett L, Antony B |title=Efficacy and Safety of Turmeric Extracts for the Treatment of Knee Osteoarthritis: a Systematic Review and Meta-analysis of Randomised Controlled Trials |journal=Curr Rheumatol Rep |volume=23 |issue=2 |pages=11 |date=January 2021 |pmid=33511486 |doi=10.1007/s11926-020-00975-8 |s2cid=231724282 |url=}}&lt;/ref&gt;
{{clear}}

==''Cynara cardunculus''==
{{main|Remedy/Plants/Asteraceae}}

==''Elaeagnus angustifolia''==
[[Image:Elaeagnus angustifolia 0353.JPG|thumb|right|250px|''Elaeagnus angustifolia'', Russian-olive is seen from the trail east from Park Lake, Dry Falls State Park, Grand Coulee, Park Lake Side Canyon. Credit: [[c:user:Wsigmund|Walter Siegmund]].{{tlx|free media}}]]
Family:	''Elaeagnaceae''.

There is evidence supporting beneficial effects of aqueous extract of Persian olive in reducing the symptoms of osteoarthritis with an efficacy comparable to that of acetaminophen and ibuprofen.&lt;ref name=Panahi&gt;{{Cite journal |pmc = 4908661|year = 2016|last1 = Panahi|first1 = Y.|title = Efficacy of Elaeagnus Angustifolia extract in the treatment of knee osteoarthritis: A randomized controlled trial|journal = Excli Journal|volume = 15|pages = 203–210|last2 = Alishiri|first2 = G. H.|last3 = Bayat|first3 = N.|last4 = Hosseini|first4 = S. M.|last5 = Sahebkar|first5 = A.|pmid = 27330526 }}&lt;/ref&gt;
{{clear}}

==''Equisetum telmateia''==
[[Image:Equisetopsida.jpg|thumb|right|200px|''Equisetum telmateia'' (''Equisetopsida'') is shown at Cambridge Botanic Garden. Credit: [[c:user:Rror|Rror]]{{tlx|free media}}]]
Family ''Equisetaceae''.

''Equisetum telmateia'', the great horsetail or northern giant horsetail, is a species with an unusual distribution, with one subspecies endemic to Europe, western Asia and northwest Africa, and a second subspecies native to western North America.&lt;ref name=grin&gt;&quot;Equisetum telmateia&quot;. Germplasm Resources Information Network (GRIN). Agricultural Research Service (ARS), United States Department of Agriculture (USDA). Retrieved 1 January 2018.&lt;/ref&gt;&lt;ref name=hwh&gt;Hyde, H. A., Wade, A. E., &amp; Harrison, S. G. (1978). ''Welsh Ferns''. National Museum of Wales ISBN:0-7200-0210-9&lt;/ref&gt;
{{clear}}

==''Eschscholzia californica''==

Family ''Papaveraceae''.

Berberine is found in ''Eschscholzia californica'' (Californian poppy).

Californidine is a chemical compound found in ''Eschscholzia californica''.

==''Eurycoma longifolia''==

Family ''Simaroubaceae''.

Among standardization markers that have been used for ''E.{{nbsp}}longifolia'' are eurycomanone, total protein, total polysaccharide and glycosaponin, which have been recommended in a technical guideline developed by the Scientific and Industrial Research Institute of Malaysia (SIRIM).&lt;ref&gt;{{cite book | title=Phytopharmaceutical Aspect Of Freeze Dried Water Extract From Tongkat Ali Roots (MS 2409:2011) | date=2011 | url=https://www.msonline.gov.my/download_file.php?file=25552&amp;source=production| publisher=Malaysia: Scientific and Industrial Research Institute of Malaysia | accessdate=2016-08-17 }}&lt;/ref&gt;

''Eurycoma longifolia'' has been reported to contain the glycoprotein compounds eurycomanol, eurycomanone, and eurycomalactone.&lt;ref&gt;{{cite journal | doi = 10.1021/np400701k| pmid = 24467387
| title = NF-κB Inhibitors from Eurycoma longifolia| journal = Journal of Natural Products| volume = 77| issue = 3| pages = 483–488| date = 2014| last1 = Tran| first1 = Thi Van Anh| last2 = Malainer| first2 = Clemens| last3 = Schwaiger| first3 = Stefan| last4 = Atanasov| first4 = Atanas G.| last5 = Heiss| first5 = Elke H.| last6 = Dirsch| first6 = Verena M.| last7 = Stuppner| first7 = Hermann }}&lt;/ref&gt;

==''Foeniculum vulgare''==
[[Image:Fennel seed.jpg|thumb|right|200px|Since the seed in the fruit is attached to the pericarp, the whole fruit is often mistakenly called &quot;seed&quot;. Credit: [[c:user:Howcheng|Howcheng]]{{tlx|free media}}]]
Family ''Apiaceae''.

Fennel is widely cultivated, both in its native range and elsewhere, for its edible, strongly flavored leaves and fruits. Its aniseed or liquorice flavor&lt;ref name=Nyerges&gt;{{cite book |last1=Nyerges |first1=Christopher |title=Foraging Wild Edible Plants of North America: More than 150 Delicious Recipes Using Nature's Edibles |date=2016 |publisher=Rowman &amp; Littlefield |isbn=978-1-4930-1499-6 |page=77 |url=https://books.google.com/books?id=RwDHCgAAQBAJ}}&lt;/ref&gt; comes from anethole, an aromatic compound also found in anise and star anise, and its taste and aroma are similar to theirs, though usually not as strong.&lt;ref name=katzer&gt;Katzer's Spice Pages: [http://gernot-katzers-spice-pages.com/engl/Foen_vul.html Fennel (''Foeniculum vulgare'' Mill.)]&lt;/ref&gt;

The aromatic character of fennel fruits derives from essential oil (volatile oils) imparting mixed aromas, including trans-anethole and estragole (resembling liquorice), fenchone (mint and camphor), limonene,&lt;ref&gt;{{Cite journal|last=Badgujar|first=Shamkant B.|last2=Patel|first2=Vainav V.|last3=Bandivdekar|first3=Atmaram H.|date=2014|title=Foeniculum vulgareMill: A Review of Its Botany, Phytochemistry, Pharmacology, Contemporary Application, and Toxicology|journal=BioMed Research International|language=en|volume=2014|pages=842674|doi=10.1155/2014/842674|issn=2314-6133|pmc=4137549|pmid=25162032}}&lt;/ref&gt; 1-octen-3-ol (mushroom).&lt;ref&gt;{{cite journal|pmid=15969523|date=2005|last1=Díaz-Maroto|first1=M. C|title=Volatile components and key odorants of fennel (''Foeniculum vulgare'' Mill.) and thyme (''Thymus vulgaris'' L.) oil extracts obtained by simultaneous distillation-extraction and supercritical fluid extraction|journal=Journal of Agricultural and Food Chemistry|volume=53|issue=13|pages=5385–9|last2=Díaz-Maroto Hidalgo|first2=I. J|last3=Sánchez-Palomo|first3=E|last4=Pérez-Coello|first4=M. S|doi=10.1021/jf050340+}}&lt;/ref&gt; Other phytochemicals found in fennel fruits include polyphenols, such as rosmarinic acid and luteolin, among others in minor content.&lt;ref&gt;{{cite journal|pmid=26471600|date=2016|last1=Uusitalo|first1=L|title=Intake of selected bioactive compounds from plant food supplements containing fennel (''Foeniculum vulgare'') among Finnish consumers|journal=Food Chemistry|volume=194|pages=619–25|last2=Salmenhaara|first2=M|last3=Isoniemi|first3=M|last4=Garcia-Alvarez|first4=A|last5=Serra-Majem|first5=L|last6=Ribas-Barba|first6=L|last7=Finglas|first7=P|last8=Plumb|first8=J|last9=Tuominen|first9=P|last10=Savela|first10=K|doi=10.1016/j.foodchem.2015.08.057}}&lt;/ref&gt;

==''Ginko biloba''==

Family ''Ginkgoaceae''.

&quot;According to some sources, the medicinal use of ginkgo dates back to 2800 B.C.… However, the first undisputed written records of ginkgo come much later… Ginkgo first appears in copies of the Shen Nung pharmacopeia around the eleventh and twelfth centuries.&quot;&lt;ref name=&quot;Crane2013&quot;/&gt;

Ginkgo has been used in traditional Chinese medicine since at least the 11th century C.E.&lt;ref name=&quot;Crane2013&quot;&gt;{{cite book |last1=Crane |first1=Peter R. |title=Ginkgo: The Tree That Time Forgot |date=2013 |publisher=Yale University Press |location=New Haven |isbn=9780300213829 |page=242 }}&lt;/ref&gt; Ginkgo seeds, leaves, and nuts have traditionally been used to treat various ailments, such as dementia, asthma, bronchitis, and kidney and bladder disorders. However, there is no conclusive evidence that ginkgo is useful for any of these conditions.&lt;ref name=&quot;nccih&quot;&gt;{{cite web |url=https://www.nccih.nih.gov/health/ginkgo| date=1 August 2020 |title=Ginkgo |publisher=National Center for Complementary and Integrative Health, US National Institutes of Health |accessdate=19 February 2021 }}&lt;/ref&gt;&lt;ref name=&quot;drugs2020&quot;&gt;{{Cite web|url=https://www.drugs.com/npp/ginkgo-biloba.html|title=Ginkgo biloba|publisher=Drugs.com|date=10 December 2020|accessdate=27 May 2021}}&lt;/ref&gt;&lt;ref name= &quot;Faran1997&quot;&gt;{{cite book |last1=Faran |first1=Mina |last2=Tcherni |first2=Anna |title=Medicinal herbs in Modern Medicine (ṣimḥei marpé bir'fū'ah ha-modernīt)|volume=1 |publisher=Akademon (Hebrew University of Jerusalem) |date=1997 |location=Jerusalem|pages=77–78 
|isbn=965-350-068-6}}, s.v. ''Ginkgo biloba''&lt;/ref&gt;

The European Medicines Agency Committee on Herbal Medicinal Products concluded that medicines containing ginkgo leaf can be used for treating mild age-related dementia and mild peripheral vascular disease in adults after serious conditions have been excluded by a physician.&lt;ref&gt;{{cite web |title=Ginkgo folium |url=https://www.ema.europa.eu/en/medicines/herbal/ginkgo-folium |publisher=European Medicines Agency |access-date=11 May 2021}}&lt;/ref&gt;

==''Glycyrrhiza glabra''==
{{main|Remedy/Plants/Fabaceae}}

==''Griffonia simplicifolia''==
{{main|Remedy/Plants/Fabaceae}}

==''Hedychium coronarium''==
[[Image:Ngải tiên.jpg|thumb|right|200px|White ginger is the Asian spice galangal. Credit: [[c:user:Bùi Thụy Đào Nguyên|Bùi Thụy Đào Nguyên]].{{tlx|free media}}]]
Family:	''Zingiberaceae''.

In China it is cultivated for use in medicine and production of aromatic oil, due to the strong characteristic fragrance of the flowers, said to be reminiscent of jasmine.&lt;ref name=Hedychium&gt;{{Cite web
|url=http://www.efloras.org/florataxon.aspx?flora_id=2&amp;taxon_id=200028392|title=''Hedychium coronarium'' in Flora of China @ efloras.org|website=www.efloras.org|accessdate=2017-02-15}}&lt;/ref&gt;&lt;ref name=Butterfly&gt;{{Cite web|url=http://www.cabi.org/isc/datasheet/26678|title=''Hedychium coronarium'' (white butterfly ginger lily)|website=www.cabi.org|accessdate=2017-02-15}}&lt;/ref&gt;
{{clear}}

==''Hibiscus acetosella''==

Family ''Malvaceae''.

In Angola a tea made from the leaves of cranberry hibiscus are used as a post-fever tonic and to treat anemia.&lt;ref name=&quot;Grubben2004&quot;/&gt; The plant is also utilized to treat myalgias by crushing leaves into cold water to bathe children.&lt;ref name=&quot;Grubben2004&quot;&gt;{{cite book|author=Grubben, Gerardus J. H. |title=Vegetables|url=https://archive.org/details/bub_gb_6jrlyOPfr24C|date=2004|publisher=PROTA|isbn=978-90-5782-147-9|pages=[https://archive.org/details/bub_gb_6jrlyOPfr24C/page/n312 312]–313}}&lt;/ref&gt;  The plant is thought to contain polyphenols, a compound that may combat inflammation and is commonly used to treat inflammatory diseases.&lt;ref name =Tsumbu&gt;{{cite journal|authors=Tsumbu CN, Deby-Dupont G, Tits M, Angenot L, Frederich M, Kohnen S, Mouithys-Mickalad A, Serteyn D, Franck T |title=Polyphenol Content and Modulatory Activities of Some Tropical Dietary Plant Extracts on the Oxidant Activities of Neutrophils and Myeloperoxidase|journal= International Journal of Molecular Sciences|year= 2012|volume= 13|issue=1|pages=628–650|doi=10.3390/ijms13010628|pmid=22312276|pmc=3269710}}&lt;/ref&gt;

==''Hibiscus cannabinus''==

Kenaf seeds yield an edible vegetable oil. The kenaf seed oil is also used for cosmetics, industrial lubricants and for biofuel production. Kenaf oil is high in omega polyunsaturated fatty acids (PUFAs). Kenaf seed oil contains a high percentage of linoleic acid (Omega-6) a polyunsaturated fatty acid (PUFA). Linoleic acid (C18:2) is the dominant PUFA, followed by oleic acid (C18:1). Alpha-linolenic acid (C18:3) is present in 2 to 4 percent.

Kenaf seed oil is 20.4% of the total seed weight, similar to that of cotton seed.

Kenaf Edible Seed Oil Contains:
*Palmitic acid: 19.1%
*Oleic acid: 28.0%  (Omega-9)
*Linoleic acid: 45% (Omega-6)
*Stearic acid: 3.0%
*Alpha-linolenic acid: 3% (Omega-3)

==''Hibiscus sabdariffa''==

The ''Hibiscus'' leaves are a good source of polyphenolic compounds. The major identified compounds include neochlorogenic acid, chlorogenic acid, cryptochlorogenic acid, caffeoylshikimic acid and flavonoid compounds such as quercetin, kaempferol and their derivatives.&lt;ref&gt;Zhen, Jing, et al. &quot;Phytochemistry, antioxidant capacity, total phenolic content and anti-inflammatory activity of Hibiscus sabdariffa leaves.&quot; Food chemistry 190 (2016): 673-680&lt;/ref&gt; The flowers are rich in anthocyanins, as well as protocatechuic acid. The dried calyces contain the flavonoids gossypetin, hibiscetine and sabdaretine. The major pigment is not daphniphylline.&lt;ref&gt;https://biologicalstaincommission.org/the-stain-extracted-from-roselle-is-not-daphniphylline/&lt;/ref&gt; Small amounts of myrtillin (delphinidin 3-monoglucoside), chrysanthenin (cyanidin 3-monoglucoside), and delphinidin are present. Roselle seeds are a good source of lipid-soluble antioxidants, particularly gamma-tocopherol.&lt;ref&gt;Mohamed R. Fernandez J. Pineda M. Aguilar M..&quot;Roselle (Hibiscus sabdariffa) seed oil is a rich source of gamma-tocopherol.&quot; ''Journal of Food Science''. 72(3):S207-11, 2007 Apr.&lt;/ref&gt;

==''Hibiscus tiliaceus''==

Cyanidin-3-glucoside is the major anthocyanin found in flowers of ''H. tiliaceus''.&lt;ref&gt;Lowry, J.B. (1976). “Floral anthocyanins of some Malesian ''Hibiscus'' species”. ''Phytochemistry'' '''15''': 1395–1396.&lt;/ref&gt; Leaves of ''H. tiliaceus'' displayed strong free radical scavenging activity and the highest tyrosinase inhibition activity among 39 tropical plant species in Okinawa.&lt;ref&gt;(Masuda ''et al''., 1999; 2005)&lt;/ref&gt; With greater UV radiation in coastal areas, it is possible that leaves and flowers of natural coastal populations of ''H. tiliaceus'' have stronger antioxidant properties than planted inland populations.&lt;ref&gt;(Wong ''et al''., 2009; Wong &amp; Chan, 2010).&lt;/ref&gt;

==''Hibiscus syriacus''==

''Hibiscus''&lt;ref&gt;''Oxford English Dictionary''&lt;/ref&gt;&lt;ref&gt;Sunset Western Garden Book, 1995:606–607&lt;/ref&gt; is a genus of flowering plants in the mallow family, ''Malvaceae''.

The generic name is derived from the Greek name ἰβίσκος (''ibískos'') which Pedanius Dioscorides gave to ''Althaea officinalis'' (ca 40–90 AD).&lt;ref&gt;{{cite book |url=https://books.google.com/books?id=VSEbamKS5uQC |title=Hibiscus: Hardy and Tropical Plants for the Garden |first=Barbara Perry |last=Lawton |publisher=Timber Press |isbn=978-0-88192-6545|year=2004 |page=36}}&lt;/ref&gt;&lt;ref&gt;
Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Di)bi%2Fskos ἰβίσκος]&lt;/ref&gt;

Several species are widely cultivated as ornamental plants, notably ''Hibiscus syriacus'' and ''Hibiscus rosa-sinensis''.&lt;ref name=RHSAZ&gt;{{cite book | editor-last = Brickell | editor-first = Christopher | title = The Royal Horticultural Society A-Z Encyclopedia of Garden Plants | year = 2008 | page = 534 | publisher = Dorling Kindersley | location = United Kingdom | isbn = 9781405332965}}&lt;/ref&gt;

==''Hippomane mancinella''==
[[Image:Hippomane mancinella (fruit).jpg|thumb|right|250px|Fruit and foliage are poisonous and shown. Credit: [[c:user:Biopics|Hans Hillewaert]].{{tlx|free media}}]]
Family: ''Euphorbiaceae''

The tree contains 12-deoxy-5-hydroxyphorbol-6-gamma-7-alpha-oxide, hippomanins, mancinellin, and sapogenin, phloracetophenone-2,4-dimethylether is present in the leaves, while the fruits possess physostigmine.&lt;ref&gt;{{Cite web
|title=''Hippomane mancinella'' |website=Dr. Duke's Phytochemical and Ethnobotanical Databases |publisher=United States Department of Agriculture
|url=https://web.archive.org/web/20041110134449/http://sun.ars-grin.gov:8080/npgspub/xsql/duke/plantdisp.xsql?taxon=475 |archive-date=2004-11-10 |accessdate=27 January 2009}}&lt;/ref&gt;

A gum can be produced from the bark which reportedly treats edema, while the dried fruits have been used as a diuretic.&lt;ref name=McLendon&gt;{{Cite web |url=http://www.mnn.com/family/protection-safety/blogs/why-manchineel-might-be-earths-most-dangerous-tree |title=Why manchineel might be Earth's most dangerous tree |last=McLendon |first=Russell |website=Mother Nature Network |publisher=Narrative Content Group |accessdate=2015-11-29}}&lt;/ref&gt;
{{clear}}

==''Huperzia serrata''==

Family ''Lycopodiaceae''.

Huperzine A is a naturally occurring sesquiterpene alkaloid compound found in the firmoss ''Huperzia serrata''&lt;ref name=&quot;Zangara2003&quot;/&gt; and in varying quantities in other food ''Huperzia'' species, including  ''H. elmeri'', ''H. carinat'', and  ''H. aqualupian''.&lt;ref&gt;{{cite journal | first1=WH | last1=Lim | last2=Goodger | first2=JQ | last3=Field | first3=AR | last4=Holtum | first4=JA | last5=Woodrow | first5=IE
| title=Huperzine alkaloids from Australasian and southeast Asian Huperzia | journal=Pharmaceutical Biology | date=2010 | volume=48 | issue=9 | pages=1073–1078 | doi=10.3109/13880209.2010.485619 | pmid=20731560 }}&lt;/ref&gt; Huperzine A has been investigated as a treatment for neurological conditions such as Alzheimer's disease, but a meta-analysis of those studies concluded that they were of poor methodological quality and the findings should be interpreted with caution.&lt;ref name=meta2013/&gt;&lt;ref name=&quot;cochranemeta&quot;&gt;{{cite journal | date=16 April 2008 | first1=J | last1=Li | last2=Wu | first2=HM | last3=Zhou | first3=RL | last4=Liu | first4=GJ | last5=Dong | first5=BR
| title=Huperzine A for Alzheimer's disease | journal=Cochrane Database of Systematic Reviews | volume=CD005592 | issue=2 | pages=CD005592 | doi=10.1002/14651858.CD005592.pub2 | pmid=18425924 }}&lt;/ref&gt;

Huperzine A is extracted from ''Huperzia serrata''.&lt;ref name=&quot;Zangara2003&quot;&gt;{{cite journal | first1=A | last1=Zangara
| title=The psychopharmacology of huperzine A: an alkaloid with cognitive enhancing and neuroprotective properties of interest in the treatment of Alzheimer's disease | journal=Pharmacology Biochemistry and Behavior | date=2003 | volume=75 | issue=3 | pages=675–686 | doi=10.1016/S0091-3057(03)00111-4 | pmid=12895686| s2cid=36435892 }}&lt;/ref&gt; &quot;Huperzine A (HupA), a novel alkaloid isolated from the Chinese herb Huperzia serrata, is a potent, highly specific and reversible inhibitor of acetylcholinesterase (AChE).&quot;&lt;ref name=Wang/&gt; It is a reversible acetylcholinesterase inhibitor&lt;ref name=Wang&gt;{{ cite journal |last1=Wang |first1=R |last2=Yan |first2=H |last3=Tang |first3=XC |date=January 2006 |title=Progress in studies of huperzine A, a natural cholinesterase inhibitor from Chinese herbal medicine |url=http://www.chinaphar.com/article/view/3574 |journal=Acta Pharmacologica Sinica |volume=27 |issue=1 |pages=1–26 |accessdate=6 December 2017
|doi=10.1111/j.1745-7254.2006.00255.x|pmid=16364207 }}&lt;/ref&gt;&lt;ref&gt;{{cite book | url=https://books.google.com/books?id=a5AMBY9ekzcC&amp;q=Huperzine&amp;pg=PA191 | title=Herbs and Nutrients for the Mind: A Guide to Natural Brain Enhancers | publisher=Greenwood Publishing Group |author1=Meletis, Chris D.  |author2=Jason E. Barke | date=2004 | pages=191 | isbn=978-0275983949}}&lt;/ref&gt;&lt;ref name=&quot;Alzheimer 1996&quot;&gt;{{cite journal | first1=BS | last1=Wang | url=https://www.scribd.com/doc/295031747/Efficacy-and-Safety-of-Natural-Acetylcholinesterase-Inhibitor-Huperzine-a-in-the-Treatment-of-Alzheimer-s-Disease-An-Updated-Meta-Analysis | title=Efficacy and safety of natural acetylcholinesterase inhibitor huperzine A in the treatment of Alzheimer's disease: an updated meta-analysis | last2=Wang | first2=H | last3=Wei | first3=ZH | last4=Song | first4=YY | last5=Zhang | first5=L | last6=Chen | first6=HZ | journal=Journal of Neural Transmission | name-list-style=vanc | date=2009 | volume=116 | issue=4 | pages=457–465 | doi=10.1007/s00702-009-0189-x | pmid=19221692| s2cid=8655284 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal | doi=10.1358/dof.1999.024.06.545143 | first1=X.C. | last1=Tang | name-list-style=vanc | title=Huperzine A: A novel acetylcholinesterase inhibitor | last2=He | first2=X.C. | last3=Bai | first3=D.L. | journal=Drugs of the Future | date=1999 | volume=24 | issue=6 | pages=647}}&lt;/ref&gt; and NMDA receptor antagonist&lt;ref&gt;{{cite journal | first1=BR | title=[+]-Huperzine A treatment protects against N-methyl-d-aspartate-induced seizure/status epilepticus in rats | first3=SA | first7=MP | journal=Chemico-Biological Interactions | last1=Coleman | last2=Ratcliffe | first2=RH | last3=Oguntayo | last4=Shi | first4=X | last5=Doctor | first5=BP | last6=Gordon | first6=RK | last7=Nambiar | name-list-style=vanc | date=2008 | volume=175 | issue=1–3 | pages=387–395 | doi=10.1016/j.cbi.2008.05.023 | pmid=18588864}}&lt;/ref&gt; that crosses the blood-brain barrier.&lt;ref&gt;{{cite journal | first1=J | last1=Patocka
| url=ftp://orbis.lfhk.cuni.cz/Acta_Medica/1998/!AM498.PDF | title=Huperzine A - an interesting anticholinesterase compound from the Chinese herbal medicine | journal=Acta Medica  | date=1998 | volume=41 | issue=4 | pages=155–7 | pmid=9951045| doi=10.14712/18059694.2019.181 }}&lt;/ref&gt; Acetylcholinesterase is an enzyme that catalyzes the breakdown of the neurotransmitter acetylcholine and of some other choline esters that function as neurotransmitters. The structure of the complex of huperzine A with acetylcholinesterase has been determined by X-ray crystallography (PDB code: [http://oca.weizmann.ac.il/oca-bin/ocaids?id=1vot 1VOT]; [http://www.proteopedia.org/wiki/index.php/1vot see the 3D structure]).&lt;ref&gt;{{cite journal | first1=ML | first3=YP | last1=Raves | title=Structure of acetylcholinesterase complexed with the nootropic alkaloid, (–)-huperzine A | last2=Harel | first2=M | last3=Pang | last4=Silman | first4=I | last5=Kozikowski | first5=AP | last6=Sussman | first6=JL | journal=Nature Structural &amp; Molecular Biology | name-list-style=vanc | date=1997 | volume=4 | issue=1 | pages=57–63 | doi=10.1038/nsb0197-57 | pmid=8989325| s2cid=236518 }}&lt;/ref&gt;

For some years, huperzine A has been investigated as a possible treatment for diseases characterized by neurodegeneration, particularly Alzheimer's disease.&lt;ref name=&quot;Zangara2003&quot; /&gt;&lt;ref name=&quot;r1&quot;&gt;{{cite journal | first1=DL | last1=Bai | last2=Tang | first2=XC | last3=He | first3=XC | name-list-style=vanc | title=Huperzine A, A Potential Therapeutic Agent for Treatment of Alzheimer's Disease | journal=Current Medicinal Chemistry | date=2000 | volume=7 | issue=3 | pages=355–374 | doi=10.2174/0929867003375281 | pmid=10637369}}&lt;/ref&gt;  A 2013 meta-analysis found that huperzine A may be efficacious in improving cognitive function, global clinical status, and activities of daily living for individuals with Alzheimer's disease. However, due to the poor size and quality of the clinical trials reviewed, huperzine A should not be recommended as a treatment for Alzheimer's disease unless further high quality studies confirm its beneficial effects.&lt;ref name=meta2013&gt;{{cite journal|name-list-style=vanc|doi=10.1371/journal.pone.0074916 | title = Huperzine A for Alzheimer's Disease: A Systematic Review and Meta-Analysis of Randomized Clinical Trials| date= 2013| last1 = Yang| first1 = Guoyan| last2 = Wang| first2 = Yuyi| last3 = Tian| first3 = Jinzhou| last4 = Liu| first4 = Jian-Ping| journal = PLOS ONE| volume = 8| issue = 9| pages = e74916| pmid = 24086396| pmc = 3781107|bibcode=2013PLoSO...874916Y }}&lt;/ref&gt;

Huperzine A is also marketed as a dietary supplement with claims made for its ability to improve memory and mental function.&lt;ref name=&quot;Talbott2012&quot;&gt;{{cite book|first=SM|last=Talbott|work=A Guide to Understanding Dietary Supplements| title=Huperzine A (HupA)|url=https://books.google.com/books?id=9ZZrW_j9XrcC&amp;pg=PA304|date= 2012|publisher=Routledge|isbn=978-1-136-80570-7|pages=304–}}&lt;/ref&gt;

Huperzine A has also been noted to help induce lucid dreaming.&lt;ref name=&quot;Ferris 2009&quot;&gt;{{cite web|title=Lucid Dreaming: A Beginner's Guide|url=http://fourhourworkweek.com/2009/09/21/how-to-lucid-dream/|website=The Four Hour Work Week|accessdate=29 December 2016}}&lt;/ref&gt;

==''Hydrastis canadensis''==

Family: ''Ranunculaceae''

Berberine is found in ''Hydrastis canadensis'' (goldenseal).&lt;ref&gt;{{cite journal | author = Zhang Q, Cai L, Zhong G, Luo W | title = Simultaneous determination of jatrorrhizine, palmatine, berberine, and obacunone in Phellodendri Amurensis Cortex by RP-HPLC | journal = Zhongguo Zhong Yao Za Zhi = Zhongguo Zhongyao Zazhi = China Journal of Chinese Materia Medica | volume = 35 | issue = 16 | pages = 2061–4 | date = 2010 | pmid = 21046728 | doi = 10.4268/cjcmm20101603 }}&lt;/ref&gt;

==''Hypericum perforatum''==
[[Image:Saint John's wort flowers.jpg|thumb|right|250px|St. John's Wort flowers are shown. Credit: [[c:user:Fir0002|Fir0002]].{{tlx|free media}}]]
Family: ''Hypericaceae''

Many members of this family contain the naphthodianthrone derivatives hypericin and pseudohypericin contained in glandular tissues that appear as black, orange or translucent spots or lines on petals, leaves and other parts of the plant, that are photosensitive and can cause reactions in grazing animals, such as blistering of the snout (muzzle), as well as in people who come into contact with the plants over prolonged periods.&lt;ref name=Robson&gt;{{ cite web |url=http://www.efloras.org/florataxon.aspx?flora_id=1&amp;taxon_id=10436 |title=Hypericaceae Jussieu: St John's Wort Family |author=Robson, Norman K.B. |publisher=Flora of North America |accessdate=17 June 2018}}&lt;/ref&gt; The highest concentration of these substances occurs in ''Hypericum perforatum'' (common St. John's wort), which is used in herbalism and as a traditional medicine (folk remedy).&lt;ref name=&quot;NCCIH&quot;&gt;{{ cite web|title=St. John's Wort|url=https://nccih.nih.gov/health/stjohnswort/ataglance.htm|publisher=National Center for Complementary and Integrative Health, US National Institutes of Health|accessdate=17 June 2018|date=September 2016 }}&lt;/ref&gt;

The plant contains the following:&lt;ref name=Barnes&gt;{{cite book |author1= Barnes, J. |author2= Anderson, L.A. |author3= Phillipson, J.D. |title= Herbal Medicines |date=2007
|publisher= Pharmaceutical Press |isbn=978-0-85369-623-0 |edition=3rd |location= London, UK |orig-year= 1996 |accessdate=7 February 2015
|url= https://web.archive.org/web/20180701140306/http://file.zums.ac.ir/ebook/366-Herbal%20Medicines,%20Third%20edition-Joanne%20Barnes%20J.%20David%20Phillipson%20Linda%20A.%20Anderson-085369623.pdf |archive-date=1 July 2018 }}&lt;/ref&gt;&lt;ref name=Greeson&gt;{{cite journal |last1=Greeson |first1=Jeffrey M. |last2=Sanford |first2=Britt |last3=Monti |first3=Daniel A. |title=St. John's wort (Hypericum perforatum): a review of the current pharmacological, toxicological, and clinical literature |journal=Psychopharmacology |date=February 2001 |volume=153 |issue=4 |pages=402–414 |doi=10.1007/s002130000625 |pmid=11243487 |s2cid=22986104 }}&lt;/ref&gt;
* Flavonoids (e.g. epigallocatechin, rutin, hyperoside, isoquercetin, quercitrin, quercetin, amentoflavone, biapigenin, astilbin, myricetin, miquelianin, kaempferol, luteolin)
* Phenolic acids (e.g. chlorogenic acid, caffeic acid, p-coumaric acid, ferulic acid, p-hydroxybenzoic acid, vanillic acid)
* Naphthodianthrones (e.g. hypericin, pseudohypericin, protohypericin, protopseudohypericin)
* Phloroglucinols (e.g. hyperforin, adhyperforin)
* Tannins (unspecified, proanthocyanidins reported)
* Volatile oils (e.g. 2-methyloctane, nonane, 2-methyldecane, undecane, α-pinene, β-pinene, α-terpineol, geraniol, myrcene, limonene, caryophyllene, humulene)
* Saturated fatty acids (e.g. isovaleric acid (3-methylbutanoic acid), myristic acid, palmitic acid, stearic acid)
* Alkanols (e.g. 1-tetracosanol, 1-hexacosanol)
* Vitamins &amp; their analogues (e.g. carotenoids, choline, nicotinamide, nicotinic acid)
* Miscellaneous others (e.g. pectin, β-sitosterol, hexadecane, triacontane, kielcorin, norathyriol)

The naphthodianthrones hypericin and pseudohypericin along with the phloroglucinol derivative hyperforin are thought to be among the numerous active constituents.&lt;ref name=&quot;Mehta&quot; /&gt;&lt;ref&gt;{{cite journal | authors = Umek A, Kreft S, Kartnig T, Heydel B | title = Quantitative phytochemical analyses of six hypericum species growing in slovenia | journal = Planta Med. | volume = 65 | issue = 4 | pages = 388–90 | year = 1999 | pmid = 17260265 | doi = 10.1055/s-2006-960798 }}&lt;/ref&gt;&lt;ref name=Tatsis&gt;{{cite journal | authors = Tatsis EC, Boeren S, Exarchou V, Troganis AN, Vervoort J, Gerothanassis IP | title = Identification of the major constituents of Hypericum perforatum by LC/SPE/NMR and/or LC/MS | journal = Phytochemistry | volume = 68 | issue = 3 | pages = 383–93 | year = 2007 | pmid = 17196625 | doi = 10.1016/j.phytochem.2006.11.026 }}&lt;/ref&gt;&lt;ref name=Schwob&gt;Schwob I, Bessière JM, Viano J.Composition of the essential oils of Hypericum perforatum L. from southeastern France.C R Biol. 2002;325:781-5.&lt;/ref&gt; It also contains essential oils composed mainly of sesquiterpenes.&lt;ref name=&quot;Mehta&quot;&gt;{{cite web|last=Mehta |first=Sweety |url=http://pharmaxchange.info/press/2012/12/pharmacognosy-of-st-johns-wort/ |title=Pharmacognosy of St. John's Wort |publisher=Pharmaxchange.info |date=2012-12-18 |accessdate=2014-02-16}}&lt;/ref&gt;

===Hyperforin===

&quot;These researches are according to an investigation of the effect of ''H. perforatum'' on the NF-κB inflammation factor, conducted by Bork et al. (1999), in which hyperforin provided a potent inhibition of TNFα-induced activation of NF-κB [58]. Another important activity for hyperforin is a dual inhibitor of cyclooxygenase-1 and 5-lipoxygenase [59]. Moreover, this species attenuated the expression of iNOS in periodontal tissue, which may contribute to the attenuation of the formation of nitrotyrosine, an indication of nitrosative stress [26]. In this context, a combination of several active constituents of Hypericum species is the carrier of their anti-inflammatory activity.&quot;&lt;ref name=Melo/&gt;

&quot;Anti-inflammatory mechanisms of hyperforin have been described as inhibition of cyclooxygenase-1 (but not COX-2) and 5-lipoxygenase at low concentrations of 0.3 μmol/L and 1.2 μmol/L, respectively [52], and of PGE2 production in vitro [53] and in vivo with superior efficiency (ED50 = 1 mg/kg) compared to indomethacin (5 mg/kg) [54]. Hyperforin turned out to be a novel type of 5-lipoxygenase inhibitor with high effectivity in vivo [55] and suppressed oxidative bursts in polymorphonuclear cells at 1.8 μmol/L in vitro [56]. Inhibition of IFN-γ production, strong downregulation of CXCR3 expression on activated T cells, and downregulation of matrix metalloproteinase 9 expression caused Cabrelle et al. [57] to test the effectivity of hyperforin in a rat model of experimental allergic encephalomyelitis (EAE). Hyperforin attenuated the symptoms significantly, and the authors discussed hyperforin as a putative therapeutic molecule for the treatment of autoimmune inflammatory diseases sustained by Th1 cells.&quot;&lt;ref name=Wolfle&gt;{{cite journal |authors=Wölfle U, Seelinger G, Schempp CM | title = Topical application of St. John's wort (Hypericum perforatum) | journal = Planta Med. | volume = 80 | issue = 2–3 | pages = 109–20 |date=February 2014  | pmid = 24214835 | doi = 10.1055/s-0033-1351019 }}&lt;/ref&gt;

&quot;Hyperforin is found in alcoholic beverages. Hyperforin is a constituent of ''Hypericum perforatum'' (St John's Wort) Hyperforin is a phytochemical produced by some of the members of the plant genus ''Hypericum'', notably ''Hypericum perforatum'' (St John's wort). The structure of hyperforin was elucidated by a research group from the Shemyakin Institute of Bio-organic Chemistry (USSR Academy of Sciences in Moscow) and published in 1975. Hyperforin is a prenylated phloroglucinol derivative. Total synthesis of hyperforin has not yet been accomplished, despite attempts by several research groups. Hyperforin has been shown to exhibit anti-inflammatory, anti-tumor, antibiotic and anti-depressant functions&lt;ref name=Hammer&gt;{{cite journal | pmid = 17696442 | doi=10.1021/jf0710074 | volume=55 | title=Inhibition of prostaglandin E(2) production by anti-inflammatory hypericum perforatum extracts and constituents in RAW264.7 Mouse Macrophage Cells | pmc=2365463 | year=2007 | author=Hammer KD, Hillwig ML, Solco AK, Dixon PM, Delate K, Murphy PA, Wurtele ES, Birt DF | journal=J Agric Food Chem | pages=7323–31}}&lt;/ref&gt;&lt;ref name=Sun&gt;{{cite journal | pmid = 21751836 | doi=10.1080/10286020.2011.584532 | volume=13 | title=In-vitro antitumor activity evaluation of hyperforin derivatives | year=2011 | author=Sun F, Liu JY, He F, Liu Z, Wang R, Wang DM, Wang YF, Yang DP | journal=J Asian Nat Prod Res | pages=688–99}}&lt;/ref&gt;&lt;ref name=Hubner&gt;{{cite journal | pmid = 12725578 | doi=10.1078/094471103321659951 | volume=10 | title=Treatment with Hypericum perforatum L. does not trigger decreased resistance in Staphylococcus aureus against antibiotics and hyperforin | year=2003 | author=Hübner AT | journal=Phytomedicine | pages=206–8}}&lt;/ref&gt;&lt;ref name=Muruganandam&gt;{{cite journal | pmid = 12018529 | volume=39 | title=Antidepressant activity of hyperforin conjugates of the St. John's wort, Hypericum perforatum Linn.: an experimental study | year=2001 | author=Muruganandam AV, Bhattacharya SK, Ghosal S | journal=Indian J Exp Biol | pages=1302–4}}&lt;/ref&gt;
# Arachidonate 5-lipoxygenase&amp;nbsp;...Specific function: Catalyzes the first step in leukotriene biosynthesis, and thereby plays a role in inflammatory processes&amp;nbsp;...
# Prostaglandin G/H synthase 1&amp;nbsp;... General function: Involved in peroxidase activity&quot;.&lt;ref name=HMDB30463&gt;{{cite encyclopedia|version=3.6|section=Enzymes|title=Hyperforin|section-url=http://www.hmdb.ca/metabolites/HMDB30463#enzymes|website=Human Metabolome Database|publisher=University of Alberta|access-date=12 December 2014|date=30 June 2013 }}&lt;/ref&gt;&lt;ref name=Melo&gt;{{cite journal |authors=de Melo MS, Quintans Jde S, Araújo AA, Duarte MC, Bonjardim LR, Nogueira PC, Moraes VR, de Araújo-Júnior JX, Ribeiro EA, Quintans-Júnior LJ | title = A systematic review for anti-inflammatory property of clusiaceae family: a preclinical approach | journal = Evid Based Complement Alternat Med | volume = 2014 | pages = 960258 | year = 2014 | pmid = 24976853 | pmc = 4058220 | doi = 10.1155/2014/960258 }}&lt;/ref&gt;&lt;ref name=Wolfle/&gt;

==''Juglans californica''==

The California black walnut is ''Juglans californica''.

==''Juglans cinerea''==

Butternuts are ''Juglans cinerea''.

==''Juglans major''==

The Arizona walnut is ''Juglans major''.

==''Juglans nigra''==

The eastern black walnut is ''Juglans nigra''

==''Juglans regia''==
[[Image:Walnuts - whole and open with halved kernel.jpg|thumb|right|250px|Walnuts are shown here whole and open with halved kernel. Credit: [[c:user:Iifar|Ivar Leidus]].{{tlx|free media}}]]
Family ''Juglandaceae''.

The Persian or English walnut is ''Juglans regia''.

Unlike most nuts that are high in monounsaturated fatty acids, walnut oil is composed largely of polyunsaturated fatty acids (72% of total fats), particularly alpha-linolenic acid (14%) and linoleic acid (58%), although it does contain oleic acid as 13% of total fats.&lt;ref name=nd1&gt;{{cite web|title=Nutrition facts: Nuts, walnuts, English dried per 100&amp;nbsp;g
|publisher=Condé Nast|accessdate=4 July 2014|url=https://web.archive.org/web/20140705192121/http://nutritiondata.self.com/facts/nut-and-seed-products/3138/2|archive-date=5 July 2014}}&lt;/ref&gt;

Walnut hulls contain diverse phytochemicals, such as polyphenols that stain hands and can cause skin irritation. Seven phenolic compounds, including ferulic acid, vanillic acid, coumaric acid, syringic acid, myricetin, and juglone were identified in walnut husks. Juglone, the predominant phenolic, was found in concentrations of 2-4% fresh weight.&lt;ref&gt;{{cite journal|last1=Cosmulescu|first1=Sina Niculina|last2=Trandafir|first2=Ion|last3=Achim|first3=Gheorghe|last4=Botu|first4=Mihai|last5=Baciu|first5=Adrian|last6=Gruia|first6=Marius|title=Phenolics of Green Husk in Mature Walnut Fruits|journal=Notulae Botanicae Horti Agrobotanici Cluj-Napoca|date=15 June 2010|volume=38|issue=1|pages=53–56|doi=10.15835/nbha3814624
|accessdate=11 October 2016|issn=1842-4309|url=https://web.archive.org/web/20170729032755/http://www.notulaebotanicae.ro/index.php/nbha/article/view/4624|archive-date=29 July 2017 }}&lt;/ref&gt;

Walnuts also contain the ellagitannin pedunculagin.&lt;ref&gt;Metabolism of Antioxidant and Chemopreventive Ellagitannins from Strawberries, Raspberries, Walnuts, and Oak-Aged Wine in Humans: Identification of Biomarkers and Individual Variability. Begoña Cerdá, Francisco A. Tomás-Barberán, and Juan Carlos Espín, J. Agric. Food Chem., 2005, 53 (2), pages 227–235, {{doi|10.1021/jf049144d}}&lt;/ref&gt; Regiolone has been isolated with juglone, betulinic acid and sitosterol from the stem bark of ''Juglans regia''.&lt;ref&gt;(−)-Regiolone, an α-tetralone from Juglans regia: structure, stereochemistry and conformation. Sunil K. Talapatra, Bimala Karmacharya, Shambhu C. De and Bani Talapatra, Phytochemistry, Volume 27, Issue 12, 1988, pages 3929–3932, {{doi|10.1016/0031-9422(88)83047-4}}&lt;/ref&gt;

==''Justicia gendarussa''==
[[Image:Justicia gendarussa 07.JPG|thumb|right|250px|''Justicia gendarussa'' flowers are shown. Credit: [[c:user:v|Vinayaraj]]{{tlx|free media}}]]
Family:	''Acanthaceae''

The plant is widely used in various forms for many of its medicinal and insecticidal properties,&lt;ref name=Agastian&gt;{{cite journal |last1=Agastian |first1=P. |last2=Williams |first2=Lincy |last3=Ignacimuthu |first3=S. |title=''In vitro'' propagation of ''Justicia gendarussa'' Burm. f.–A medicinal plant |journal=Indian Journal of Biotechnology |date=April 2006 |volume=5 |issue=2 |pages=246–248 |url=http://nopr.niscair.res.in/handle/123456789/7756 |issn=0975-0967}}&lt;/ref&gt;

''Justicia gendarussa'' is harvested for its leaves for the treatment of various ailments.&lt;ref name=Dennis&gt;{{Cite journal|last1=Dennis Thomas|first1=T.|last2=Yoichiro|first2=Hoshino|date=September 2010|title=''In vitro'' propagation for the conservation of a rare medicinal plant ''Justicia gendarussa'' Burm. f. by nodal explants and shoot regeneration from callus|url=http://link.springer.com/10.1007/s11738-010-0482-1|journal=Acta Physiologiae Plantarum
|volume=32|issue=5|pages=943–950|doi=10.1007/s11738-010-0482-1|s2cid=24975195|issn=0137-5881}}&lt;/ref&gt;

It maybe useful for the treatment of asthma, rheumatism and colics of children.&lt;ref&gt;medicinal uses [https://archive.org/stream/pharmacographia03dymogoog#page/n64/mode/1up/search/asthma pharmacographica indica]&lt;/ref&gt; It may have the potential to be the basis for a birth control pill for men. Clinical tests are being conducted in Indonesia.&lt;ref name=Winn&gt;{{cite news |url=http://www.globalpost.com/dispatch/news/asia-pacific/indonesia/110224/indonesia-birth-control-pill-papua-men |title=Indonesia's birth control pill for men |author=Patrick Winn |date=February 27, 2011 |accessdate=March 2, 2011 |publisher=GlobalPost}}&lt;/ref&gt;&lt;ref&gt;[https://www.pbs.org/newshour/bb/health/july-dec11/birth_07-18.html Indonesian Plant Shows Promise for Male Birth Control] ''PBS NewsHour'', July 20, 2011&lt;/ref&gt;&lt;ref&gt;{{cite news|title=Indonesia is about to start producing a male birth control pill that will change the world|url=http://jakarta.coconuts.co/2014/11/24/indonesia-about-start-producing-male-birth-control-pill-going-change-world|accessdate=3 February 2015|work=Coconuts Jakarta|date=24 November 2014}}&lt;/ref&gt;

The plant has shown promise as a source of a compound that inhibits an enzyme crucial to the development of HIV.&lt;ref name=Zhang&gt;{{Cite journal|doi=10.1021/acs.jnatprod.7b00004|title=Potent Inhibitor of Drug-Resistant HIV-1 Strains Identified from the Medicinal Plant ''Justicia gendarussa''|daye=2017|last1=Zhang|first1=Hong-Jie|last2=Rumschlag-Booms|first2=Emily|last3=Guan|first3=Yi-Fu|last4=Wang|first4=Dong-Ying|last5=Liu|first5=Kang-Lun|last6=Li|first6=Wan-Fei|last7=Nguyen|first7=Van H.|last8=Cuong|first8=Nguyen M.|last9=Soejarto|first9=Djaja D.|last10=Fong|first10=Harry H. S.|last11=Rong|first11=Lijun|journal=Journal of Natural Products|volume=80|issue=6|pages=1798–1807|pmid=28613071}}&lt;/ref&gt;&lt;ref&gt;[http://www.labmanager.com/news/2017/06/plant-compound-more-powerful-than-azt-against-hiv?#.WUqKcilLc2w Labmanager /2017/06/ plant compound more powerful than azt]&lt;/ref&gt;

''Justicia gendarussa'' was proved to contain several phytochemicals, which are natural secondary plant compounds. Overall in the plant, roots, stem and leaves, following phytochemicals were found: alkaloids, flavonoids, tannins and phenols.&lt;ref name=Pakistan&gt;{{Cite web|title=Pakistan Journal of Botany|url=http://pakbs.org/pjbot/paper_details.php?id=9972|accessdate=2021-12-04|website=pakbs.org}}&lt;/ref&gt; The ingredients of the plant may vary depending on the age, physiological stage of the organ parts or the geographic region of cultivation.&lt;ref name=Widyowati&gt;{{Cite journal|last1=Widyowati|first1=Retno|last2=Agil|first2=Mangestuti|date=2018|title=Chemical Constituents and Bioactivities of Several Indonesian Plants Typically Used in Jamu|url=https://www.jstage.jst.go.jp/article/cpb/66/5/66_c17-00983/_article|journal=Chemical and Pharmaceutical Bulletin|volume=66|issue=5|pages=506–518|doi=10.1248/cpb.c17-00983|pmid=29710047}}&lt;/ref&gt;

The plant was proved to have both anti-microbial and anti-fungal action on selected pathogen strains, and therefore this plant can be used to develop herbal drugs.&lt;ref name=Pakistan/&gt;

''Justicia gendarussa'' leaf extract was proven to potentially become a male, non-hormonally contraceptive method due to its competitive and reversible inhibition of the spermatozoan hyaluronidase enzyme. The plant is already used as traditional contraceptive method in Indonesia.&lt;ref name=Ratih&gt;{{Cite journal|last1=Ratih|first1=Gusti Ayu Made|last2=Imawati|first2=Maria Fatmadewi|last3=Purwanti|first3=Diah Intan|last4=Nugroho|first4=Rendra Rizki|last5=Wongso|first5=Suwidji|last6=Prajogo|first6=Bambang|last7=Indrayanto|first7=Gunawan|date=2019-06-01|title=Metabolite Profiling of Justicia gendarussa Herbal Drug Preparations|url=https://doi.org/10.1177/1934578X19856252|journal=Natural Product Communications|language=en|volume=14|issue=6|pages=1934578X19856252|doi=10.1177/1934578X19856252|s2cid=195427031|issn=1934-578X}}&lt;/ref&gt;

The plant compound Patentiflorin A contained in ''Justicia gendarussa'' has shown to have a positive activity against several HIV strains, higher than the clinically used first anti-HIV drug, zidovudine AZT.&lt;ref name=Zhang/&gt;

Further, extracts of the leaves have an anti-inflammatory effect. This has been demonstrated especially in mice, specific for the carrageenan-induced paw edema.&lt;ref name=Kavitha&gt;{{Cite journal|last1=Kavitha|first1=S. K.|last2=Viji|first2=V.|last3=Kripa|first3=K.|last4=Helen|first4=A.|date=2011-07-01|title=Protective effect of Justicia gendarussa Burm.f. on carrageenan-induced inflammation|url=https://doi.org/10.1007/s11418-011-0524-z|journal=Journal of Natural Medicines|language=en|volume=65|issue=3|pages=471–479|doi=10.1007/s11418-011-0524-z|pmid=21416126|s2cid=1271250|issn=1861-0293}}&lt;/ref&gt;

The juice of the leaves can be drizzled into the ear for earache. To treat external edema, an oil made from the leaves can be used.&lt;ref&gt;{{Cite journal|last1=Aye|first1=Mya Mu|last2=Aung|first2=Hnin Thanda|last3=Sein|first3=Myint Myint|last4=Armijos|first4=Chabaco|date=January 2019|title=A Review on the Phytochemistry, Medicinal Properties and Pharmacological Activities of 15 Selected Myanmar Medicinal Plants|journal=Molecules|volume=24|issue=2|pages=293|doi=10.3390/molecules24020293|pmid=30650546|pmc=6359042 }}&lt;/ref&gt;
{{clear}}

==''Kunzea ericoides''==

Family: ''Myrtaceae''

==''Lagerstroemia speciosa''==

Family: ''Lythraceae''

==''Leptospermum polygalifolium''==

Family: ''Myrtaceae''

The nectar from the flowers is harvested by bees, yielding Leptospermum honey, which is marketed as Manuka honey.&lt;ref name=&quot;NZMPI&quot;&gt;{{cite web |title=Growing and harvesting Mānuka honey |url=https://www.mpi.govt.nz/growing-and-harvesting/honey-and-bees/manuka-honey/ |publisher=New Zealand Ministry for Primary Industries |accessdate=5 December 2019}}&lt;/ref&gt; Honey produced from Australian ''Leptospermum polygalifolium'' is also known as ''jelly bush'' or the ''lemon-scented tea tree''.&lt;ref&gt;{{cite journal |title=Native honey a sweet antibacterial
|url=https://web.archive.org/web/20110306093531/http://www.australiangeographic.com.au/journal/native-honey-a-sweet-antibacterial.htm |date=2011-03-06 |journal=Australian Geographic }}&lt;/ref&gt;

==''Limnophila aromatica''==

Family:	''Plantaginaceae''.

==''Linum bienne''==

Family:	''Linaceae''.

==''Linum usitatissimum''==
[[Image:Flax flowers.jpg|thumb|right|250px|Flowers of Flax (''Linum usitatissimum'') are from Ottawa, Ontario, Canada. Credit: [[c:user:Dger|D. Gordon E. Robertson]].{{tlx|free media}}]]
Family:	''Linaceae''.

Flax seeds are 7% water, 18% protein, 29% carbohydrates, and 42% fat. In {{convert|100|g}} as a reference amount, flax seeds provide 534 calories and contain high levels (20% or more of the Daily Value, DV) of protein, dietary fiber, several B vitamins, and dietary minerals.&lt;ref name=nd/&gt;&lt;ref name=usda/&gt; Flax seeds are especially rich in thiamine, magnesium, and phosphorus (DVs above 90%).

As a percentage of total fat, flax seeds contain 54% omega-3 fatty acids (mostly alpha-Linolenic acid (ALA), 18% omega-9 fatty acids (oleic acid), and 6% omega-6 fatty acids (linoleic acid); the seeds contain 9% saturated fat, including 5% as palmitic acid.&lt;ref name=&quot;nd&quot;&gt;{{cite web
|title=Nutrition facts for 100 g of flaxseeds|publisher=Conde Nast for USDA National Nutrient Database, version SR-21|date=2015
|url=https://web.archive.org/web/20101205075853/http://nutritiondata.self.com/facts/nut-and-seed-products/3163/2|archive-date=2010-12-05}}&lt;/ref&gt;&lt;ref name=&quot;usda&quot;&gt;{{cite web
|title=Full Report (All Nutrients): 12220, Seeds, flaxseed per 100 g|publisher=USDA National Nutrient Database version SR-27|date=2015
|url=https://web.archive.org/web/20140920222846/http://ndb.nal.usda.gov/ndb/foods/show/3745?fg=&amp;man=&amp;lfacet=&amp;count=&amp;max=&amp;sort=&amp;qlookup=&amp;offset=&amp;format=Full&amp;new=&amp;measureby=|archive-date=2014-09-20 }}&lt;/ref&gt; Flax seed oil contains 53% 18:3 omega-3 fatty acids (mostly ALA) and 13% 18:2 omega-6 fatty acids.&lt;ref name=nd/&gt;
{{clear}}

==''Lithospermum officinale''==
[[Image:Lithospermum officinale.jpeg|thumb|right|250px|The image shows buds on ''Lithospermum officinale''. Credit: [[c:user:Fabelfroh|Kristian Peters]].{{tlx|free media}}]]
''Lithospermum officinale'', or common gromwell or European stoneseed, is a flowering plant species in the family ''Boraginaceae'', native to Eurasia.

The plant has been found to be a potent natural anti-inflammatory and effective agent for healing burn wounds when applied topically, which explains the presence of this species in the poultice discovered.&lt;ref name=Amiri&gt;Amiri ZM, Tanideh N, Seddighi A, Mokhtari M, Amini M, Partovi AS, Manafi A, Hashemi SS, Mehrabani D. 2017. The effect of ''Lithospermum officinale'', silver sulfadiazine and alpha ointments in healing of burn wound injuries in rat. ''World J Plast Surg'' 6(3): 313–318.&lt;/ref&gt;

The mechanism by which freeze-dried aqueous extracts (FDE) of plants of the species Lithospermum officinale (Boraginaceae) have the ability to inhibit at least many of the effects of exogenous and endogenous TSH on the thyroid gland. To this end, we have examined the in vitro effects of FDE from these plants on the ability of bovine TSH (bTSH) to both bind to human thyroid plasma membranes (TPM) and activate adenylate cyclase therein. FDE of this species produced a dose-related, ultimately complete, inhibition of the binding of 125I-labeled bTSH when studied at 4 C in a 20 mM Tris-HCl-0.5% BSA buffer, pH 7.45. Half-maximum inhibition of bTSH binding was produced by approximately 50 mU/ml bTSH and only about 10-30 micrograms/ml of the four active FDE. When studied in Tris-BSA-50 mM NaCl buffer at 37 C, these FDE remained inhibitory to bTSH binding, but their potency was decreased to about one fifth of that seen in the absence of NaCl. The binding of [125I]hCG to rat testis membranes was also inhibited by all of these FDE, but no effect on the binding of [125I]insulin to crude rat liver membranes was observed.

The antithyrotropic activity of freeze-dried-extracts from ''Lithospermum officinale'' (Lith. off. FDE) was investigated in the rat. When administered together with TSH, Lith. off. FDE blocked the TSH- induced increase in endocytotic activity of the thyroid glands followed by a strong decline of thyroid hormone levels. Furthermore, when Lith. off. FDE was injected alone it caused a decline in endogenous TSH-levels as well as in thyroidal secretion and thyroid hormone levels. The efficacy of the extract in blocking thyroid secretion was compared to that of potassium iodide and it was found that the effect of Lith. off. FDE was of more rapid onset and of longer duration, suggesting that the FDE may have a different mode of action from that of KJ. A specific interaction between TSH and the active constituents of the plant extract is discussed. Experiments on thyroidectomized and T4 substituted rats have demonstrated as an additional pharmacodynamic effect of Lith. off. FDE an inhibition of peripheral T4-deiodination.

''Lithospermum officinale'' has been studied using female Wistar rats. Variations of the main urolithiasis risk factors (citraturia, calciuria, phosphaturia, pH and diuresis) have been evaluated. It can be concluded that beneficial effects caused by a herb infusion on urolithiasis (kidney stones) can be attributed to some disinfectant action, and tentatively to the presence of saponins. Specifically, some solvent action can be postulated with respect to uric stones or heterogeneous uric nucleus, due to the basifying capacity of some herb infusions.
{{clear}}

==''Lonicera japonica''==
[[Image:Lonicera japonica, Fruit.JPG|thumb|right|200px|''Lonicera japonica'' fruit is growing in the Aizu area, Fukushima pref., Japan. Credit: Qwert1234.{{tlx|free media}}]]
Family: ''Caprifoliaceae''

In traditional Chinese medicine,&lt;ref name=&quot;Shang2011&quot;&gt;{{cite journal |author1=Shang, X. |author2=Pan, H. |author3=Li, M. |author4=Miao, X. |author5=Ding, H. |date=2011 |title=''Lonicera japonica'' Thunb.: Ethnopharmacology, phytochemistry and pharmacology of an important traditional Chinese medicine |journal=Journal of Ethnopharmacology |volume=138 |issue=1 |pages=1–21 |doi=10.1016/j.jep.2011.08.016 |pmid=21864666|pmc=7127058 }}&lt;/ref&gt; ''Lonicera japonica'' is called ''rěn dōng téng''&lt;ref name=&quot;Shang2011&quot; /&gt; literally &quot;winter enduring vine&quot;) or ''jīn yín huā''&lt;ref name=&quot;Shang2011&quot; /&gt; literally &quot;gold-silver flower&quot;. Alternative Chinese names include ''er hua'' and ''shuang hua'', meaning ''double-[color] flowers''.&lt;ref&gt;''Chinese Medical Herbology and Pharmacology'', John and Tina Chen, Art of Medicine Press, 1st ed. 2001, p. 171&lt;/ref&gt; In Korean, it is called ''geumeunhwa''.

The dried leaves and flowers (Flos Lonicerae Japonicae) are employed in traditional Chinese medicine, being used to treat fever, cold-related headache, cough, thirst, certain inflammation including sore throat, skin infection, and tumor necrosis.&lt;ref&gt;{{cite book |last1=Bensky |first1=Dan |last2=Barolet |first2=Randall |title=Chinese Herbal Medicine Formulas &amp; Strategies |edition=2nd |publisher=Eastland Press |page=44 }}&lt;/ref&gt;

The antiviral action of loniflavone, a compound found in ''Lonicera japonica'', has been investigated in computational studies, in which the ability of this compound to bind with high affinity to the spike protein of SARS-CoV-2 has been demonstrated, an early step towards drug development for the disease that virus causes.&lt;ref&gt;{{cite journal |last1=Kadioglu |first1=Onat |last2=Saeed |first2=Mohamed |last3=Greten |first3=Henry Johannes |last4=Efferth |first4=Thomas |title=Identification of novel compounds against three targets of SARS CoV-2 coronavirus by combined virtual screening and supervised machine learning |journal=Computers in Biology and Medicine |date=June 2021 |volume=133 |pages=104359 |doi=10.1016/j.compbiomed.2021.104359|pmc=8008812 }}&lt;/ref&gt;

''Lonicera japonica'' contains methyl caffeate, 3,4-di-O-caffeoylquinic acid, methyl 3,4-di-O-caffeoylquinate, protocatechuic acid, methyl chlorogenic acid, and luteolin. The two biflavonoids, 3′-O-methyl loniflavone and loniflavone, along with luteolin and chrysin, can be isolated from the leaves.&lt;ref&gt;{{cite journal | doi=10.1016/j.phytochem.2005.10.002| pmid=16293275| title=Biflavonoids from ''Lonicera japonica''| journal=Phytochemistry| volume=66| issue=23| pages=2740–4| year=2005| last1=Kumar| first1=Neeraj| last2=Singh| first2=Bikram| last3=Bhandari| first3=Pamita| last4=Gupta| first4=Ajai P.| last5=Uniyal| first5=Sanjay K.| last6=Kaul| first6=Vijay K.}}&lt;/ref&gt; Other phenolic compounds present in the plant are hyperoside, chlorogenic acid, and caffeic acid.&lt;ref&gt;{{cite journal | doi=10.1002/elan.200403102| title=Determination of Phenolic Acids and Flavones in ''Lonicera'' japonica Thumb. By Capillary Electrophoresis with Electrochemical Detection| journal=Electroanalysis| volume=17| issue=4| pages=356| year=2005| last1=Peng| first1=Youyuan| last2=Liu| first2=Fanghua| last3=Ye| first3=Jiannong}}&lt;/ref&gt; The two secoiridoid glycosides, loniceracetalide A and loniceracetalide B, can be isolated, together with 10 known iridoid glycosides, from the flower buds.&lt;ref&gt;{{cite journal | doi=10.1016/S0031-9422(00)00279-X| pmid=11140518| title=Secoiridoid glycosides from the flower buds of Lonicera japonica| journal=Phytochemistry| volume=55| issue=8| pages=879–81| date=2000| last1=Kakuda| first1=Rie| last2=Imai| first2=Mio| last3=Yaoita| first3=Yasunori| last4=Machida| first4=Koichi| last5=Kikuchi| first5=Masao}}&lt;/ref&gt; The plant also contains the saponins loniceroside A and loniceroside B&lt;ref&gt;{{cite journal | doi=10.1016/S0031-9422(00)90656-3| pmid=7764625| title=Triterpenoid saponins from the aerial parts of Lonicera japonica| journal=Phytochemistry| volume=35| issue=4| pages=1005–8| year=1994| last1=Ho Son| first1=Kun| last2=Young Jung| first2=Keun| last3=Wook Chang| first3=Hyeun| last4=Pyo Kim| first4=Hyun| last5=Sik Kang| first5=Sam}}&lt;/ref&gt; and the antiinflammatory loniceroside C.&lt;ref&gt;{{cite journal | doi=10.1248/cpb.51.333| pmid=12612424| title=Loniceroside C, an Antiinflammatory Saponin from Lonicera japonica| journal=Chemical &amp; Pharmaceutical Bulletin| volume=51| issue=3| pages=333–5| date=2003| last1=Kwak| first1=Wie Jong| last2=Han| first2=Chang Kyun| last3=Chang| first3=Hyeun Wook| last4=Kim| first4=Hyun Pyo| last5=Kang| first5=Sam Sik| last6=Son| first6=Kun Ho| }}&lt;/ref&gt;
{{clear}}

==''Lophatherum gracile''==

Family: ''Poaceae''

==''Lycium barbarum''==

Family: ''Solanaceae''

Because of its claimed benefits as a drug of traditional medicine, the chemicals present in the fruit, root, and other parts of the plants have been studied in some detail.&lt;ref name=potterat&gt;Olivier Potterat (2010): &quot;Goji (''Lycium barbarum'' and ''L. chinense''): Phytochemistry, pharmacology and safety in the perspective of traditional uses and recent popularity&quot;.  ''Planta medica'', volume 76, issue 1, pages 7-19.  doi=10.1055/s-0029-1186218&lt;/ref&gt;&lt;ref name=bonturi&gt;Loraine Bonturi (2015), &quot;[https://core.ac.uk/download/pdf/79618336.pdf Attività farmacologiche e possibili bersagli molecolari dei polisaccaridi del Lycium barbarum (LBP)]&quot;  Graduation Thesis, Pharmacy Department, University of Pisa. Accessed on 2018-06-12.&lt;/ref&gt;

The main compounds in the fruit (23% of the dry mass) are polysaccharides and proteoglycans.  Carotenoid pigments are the second major group, chiefly zeaxanthin palmitic acid (dipalmitate). The fruits further contain vitamins, in particular riboflavin, thiamin and ascorbic acid (vitamin C), the latter in a concentration similar to that in lemons. Other detected compounds include flavonoids derived from myricetin, quercetin, and kaempferol; hexadecanoic acid, linoleic acid, β-elemene, myristic acid and ethyl hexadecanoate;  and some glycerogalactolipids. The fruit further contains 1–2.7% of free amino acids; chiefly proline, and including gamma-aminobutyric acid (GABA) and trimethylglycine (betaine). Other compounds include β-sitosterol, scopoletin, p-coumaric acid, lyciumide A and L-monomenthyl succinate. The alkaloid atropine, common in plants of the family ''Solanaceae'', is not detectable.&lt;ref name=potterat/&gt;

The compounds present in the roots have been less studied, but they include trimethylglycine (betaine), choline, linoleic acid, and β-sitosterol [79]. Of particular interest are cyclic oligopeptides with 8 amino acid rings, christened lyciumin A and lyciumin B.&lt;ref name=potterat/&gt;

The leaves are known to contain the flavonoids quercetin 3-O-rutinoside-7-O-glucoside, kaempferol 3-O-rutinoside-7-O-glucoside, rutin, nicotiflorin, isoquercitrin, quercetin, kaempferol damascenone, choline, scopoletin, vanillic acid, salicylic acid, and nicotinic acid. From the flowers, diosgenin, β-sitosterol, and lanosterol have been isolated.&lt;ref name=potterat/&gt;

==''Lycium europaeum''==

''Lycium europaeum'', the European tea tree, European box-thorn, or European matrimony-vine, is a species of flowering plant in the family ''Solanaceae''.&lt;ref&gt;{{cite web |url=https://www.letsplant.org/lycium-europaeum |title=''Lycium europaeum'' |author= |date=2021 |website=letsplant.org |publisher=Let's Plant |accessdate=4 August 2021 }}&lt;/ref&gt; It is native to the entire Mediterranean region, and has been introduced to the Canary Islands, Madeira, and the Balearic Islands.&lt;ref name=&quot;816435-1&quot; &gt;{{cite web |url=http://powo.science.kew.org/taxon/urn:lsid:ipni.org:names:816435-1 |title=''Lycium europaeum'' L. |author= |website=Plants of the World Online |publisher=Board of Trustees of the Royal Botanic Gardens, Kew |accessdate=4 August 2021 }}&lt;/ref&gt; Its fruit is edible.&lt;ref name = &quot;1_of_1060_Google_Scholar_Hits&quot;&gt;{{cite journal |title=Phytochemical composition and health properties of ''Lycium europaeum'' L.: A review |date=2020 |last1=Aidi Wannes |first1=Wissem |last2=Saidani Tounsi |first2=Moufida |journal=Acta Ecologica Sinica |doi=10.1016/j.chnaes.2020.09.008 }}&lt;/ref&gt;

==''Magnolia grandiflora''==

Family: ''Magnoliaceae''.

==''Magnolia officinalis''==

The aromatic bark contains magnolol, honokiol, 4-O-methylhonokiol, and obovatol.&lt;ref&gt;{{cite journal |author1=Han, H. |author2=Jung, J.K. |author3=Han, S.B. |author4=Nam, S.Y. |author5=Oh, K.W. |author6=Hong, J.T. |title=Anxiolytic-like effects of 4-O-methylhonokiol isolated from magnolia officinalis through enhancement of GABAergic transmission and chloride influx |journal=Journal of Medicinal Food |volume=14 |issue=7–8 |pages=724–731 |date=2011 |doi=10.1089/jmf.2010.1111 |pmid=21501091}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author1=Kalman, D.S. |author2=Feldman, S. |author3=Feldman, R. |author4=Schwartz, H.I. |author5=Krieger, D.R. |author6=Garrison, R. |title=Effect of a proprietary Magnolia and Phellodendron extract on stress levels in healthy women: A pilot, double-blind, placebo-controlled clinical trial |journal=Nutrition Journal |volume=7 |issue=1 |date=2008 |doi=10.1186/1475-2891-7-11 |pmid=18426577 |pmc=2359758 |pages=11}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author1=Ma, L. |author2=Chen, J. |author3=Wang, X. |author4=Liang, X. |author5=Luo, Y. |author6=Zhu, W. |author7=Wang, T. |author8=Peng, M. |author9=Li, S. |author10=Jie, S. |author11=Peng, A. |author12=Wei, Y. |author13=Chen, L. |title=Structural modification of honokiol, a biphenyl occurring in magnolia officinalis: The evaluation of honokiol analogues as inhibitors of angiogenesis and for their cytotoxicity and structure-activity relationship |journal=Journal of Medicinal Chemistry |volume=54 |issue=19 |pages=6469–6481 |date=2011 |doi=10.1021/jm200830u |pmid=21853991}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author1=Fried, L.E. |author2=Arbiser, J.L. |title=Honokiol, a multifunctional antiangiogenic and antitumor agent |journal=Antioxidants &amp; Redox Signaling |volume=11 |pages=1139–1148 |date=2009 |issue=5 |doi=10.1089/ars.2009.2440 |pmid=19203212 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author1=Hu J. |author2=Chen L.-J. |author3=Liu L. |author4=Chen X. |author5=Chen P. |author6=Yang G.-L. |author7=Hou W.-L. |author8=Tang M.-H. |author9=Zhang F. |author10=Wang X.-H. |author11=Zhao X. |author12=Wei Y.-Q. |title=Liposomal honokiol, a potent anti-angiogenesis agent, in combination with radiotherapy produces a synergistic antitumor efficacy without increasing toxicity |journal=Experimental &amp; Molecular Medicine |volume=40 |issue=6 |pages=617–628 |date=2008 |doi=10.3858/emm.2008.40.6.617 |pmid=19116447 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal |author=Lee YJ, Lee YM, Lee CK, Jung JK, Han SB, Hong JT |title=Therapeutic applications of compounds in the Magnolia family |journal=Pharmacol. Ther. |date=2011 |volume=130 |issue=2 |pages=157–176 |doi=10.1016/j.pharmthera.2011.01.010 |pmid=21277893}}&lt;/ref&gt; Magnolol&lt;ref&gt;{{cite journal |last1=Fakhrudin |first1=N. |last2=Ladurner |first2=A. |last3=Atanasov |first3=A.G. |last4=Heiss |first4=E.H. |last5=Baumgartner |first5=L. |last6=Markt |first6=P. |last7=Schuster |first7=D. |last8=Ellmerer |first8=E.P. |last9=Wolber |first9=G. |last10=Rollinger |first10=J.M. |last11=Stuppner |first11=H. |last12=Dirsch |first12=V.M. |date=April 2010 |title=Computer-aided discovery, validation, and mechanistic characterization of novel neolignan activators of peroxisome proliferator-activated receptor gamma |journal=Mol. Pharmacol. |volume=77 |issue=4 |pages=559–66 |doi=10.1124/mol.109.062141 |pmid = 20064974 }}&lt;/ref&gt; and honokiol&lt;ref&gt;{{cite journal |author=Atanasov AG, Wang JN, Gu SP, Bu J, Kramer MP, Baumgartner L, Fakhrudin N, Ladurner A, Malainer C, Vuorinen A, Noha SM, Schwaiger S, Rollinger JM, Schuster D, Stuppner H, Dirsch VM, Heiss EH |title=Honokiol: A non-adipogenic PPARγ agonist from nature |pmid=23811337 |doi=10.1016/j.bbagen.2013.06.021 |volume=1830 |issue=10 |pmc=3790966 |date=October 2013 |pages=4813–4819 |journal= Biochimica et Biophysica Acta (BBA) - General Subjects}}&lt;/ref&gt; activate the nuclear receptor peroxisome proliferator-activated receptor gamma.

Magnolol is an organic compound, classified as lignan, a bioactive compound found in the bark of the Houpu magnolia (''Magnolia officinalis'') or in ''Magnolia grandiflora''.&lt;ref&gt;{{cite journal |pmid=21277893 |doi=10.1016/j.pharmthera.2011.01.010|title=Therapeutic applications of compounds in the ''Magnolia'' family|journal=Pharmacology &amp; Therapeutics|volume=130|issue=2|pages=157–76|date=2011|last1=Lee|first1=Young-Jung|last2=Lee|first2=Yoot Mo|last3=Lee|first3=Chong-Kil|last4=Jung|first4=Jae Kyung|last5=Han|first5=Sang Bae|last6=Hong|first6=Jin Tae}}&lt;/ref&gt;

==''Mahonia aquifolium''==

Family: ''Berberidaceae''.

Berberine is found in ''Mahonia aquifolium'' (Oregon grape).&lt;ref&gt;{{cite journal | author = Zhang Q, Cai L, Zhong G, Luo W | title = Simultaneous determination of jatrorrhizine, palmatine, berberine, and obacunone in Phellodendri Amurensis Cortex by RP-HPLC | journal = Zhongguo Zhong Yao Za Zhi = Zhongguo Zhongyao Zazhi = China Journal of Chinese Materia Medica | volume = 35 | issue = 16 | pages = 2061–4 | date = 2010 | pmid = 21046728 | doi = 10.4268/cjcmm20101603 }}&lt;/ref&gt;

==''Marchantia berteroana''==

Family: ''Marchantiaceae''.

Hypolaetin is a flavone, the aglycone of hypolaetin 8-glucuronide (H-8-G), a compound found in the liverwort ''Marchantia berteroana''.&lt;ref name=Markham&gt;Isoscutellarein and hypolaetin 8-glucuronides from the liverwort Marchantia berteroana. Kenneth R. Markham and Lawrence J. Porter, Phytochemistry, April 1975, Volume 14, Issue 4, Pages 1093–1097, doi:10.1016/0031-9422(75)85194-6 &lt;/ref&gt;

The &quot;effects of H-8-G and its aglycone, hypolaetin (H) on rabbit skin edema [...] Edema formation was measured as the local accumulation during 1 h of 1251-human serum albumin (5 1.tCi) (4). [...] Local administration of H-8-G or H did not influence skin edema&quot;.&lt;ref name=Junginger&gt;{{ cite journal
|author=M Junginger, M Wichtl
|title=New Cardenolide Glycosides and Adenosine from Adonis vernalis
|journal=Planta Medica
|date=1989
|volume=55
|issue=1
|pages=107
|url=https://www.thieme-connect.com/products/ejournals/abstract/10.1055/s-2006-961877
|arxiv=
|bibcode=
|doi=10.1055/s-2006-961877
|pmid=
|accessdate=14 February 2022 }}&lt;/ref&gt;

==''Matricaria recutita''==
{{main|Remedy/Plants/Asteraceae}}

==''Melaleuca alternifolia''==

Family: ''Myrtaceae''.

''Melaleuca alternifolia'' is notable for its essential oil (tea tree oil) which is both an antifungal medication and antibiotic,&lt;ref name=ncbi&gt;{{cite journal|last1=Carson|first1=C. F.|last2=Hammer|first2=K. A.|last3=Riley|first3=T. V.|title=''Melaleuca alternifolia'' (Tea Tree) Oil: a Review of Antimicrobial and Other Medicinal Properties|journal=Clinical Microbiology Reviews|date=17 January 2006|volume=19|issue=1|pages=50–62|doi=10.1128/CMR.19.1.50-62.2006|pmid=16418522|pmc=1360273}}&lt;/ref&gt; while safely usable for topical applications.&lt;ref name=RIRDC&gt;{{cite book|last1=O'Brien|first1=Peter|last2=Dougherty|first2=Tony|title=The effectiveness and safety of Australian Tea Tree oil|date=2007|publisher=RIRDC|location=Barton, A.C.T.|isbn=978-1741515398|pages=9–12|url=http://www.gelair.com.au/pdf/RIRDC-Paper-Efficacy-and-Toxicity.pdf|accessdate=19 August 2015}}&lt;/ref&gt; This is produced on a commercial scale and marketed as tea tree oil.&lt;ref name=&quot;Brophy (2)&quot;&gt;{{cite web|last1=Brophy|first1=Joseph J.|last2=Craven|first2=Lyndley A.|last3=Doran|first3=John C.|title=Melaleuca - Their Botany, Essential Oil and uses (Preliminaries)
|publisher=Australian Centre for International Agricultural Research|accessdate=19 August 2015|url=https://web.archive.org/web/20150528213958/http://aciar.gov.au/files/mn156-prelims_1.pdf|archive-date=28 May 2015 }}&lt;/ref&gt;

==''Melaleuca cajuputi''==

Family: ''Myrtaceae''.

''Melaleuca cajuputi'' is used to produce a similar oil, known as cajuput oil, which is used in Southeast Asia to treat a variety of infections and to add fragrance to food and soaps.&lt;ref name=Doran&gt;{{cite book|last1=Doran|first1=John C.|editor-last1=Southwell|editor-first1=Ian|editor-last2=Lowe|editor-first2=Robert|title=Tea tree: the genus melaleuca|date=1999|publisher=Harwood Academic|location=Amsterdam|isbn=9057024179|pages=221–224}}&lt;/ref&gt;

==''Mentha × piperita''==
[[Image:Pfefferminze natur peppermint.jpg|thumb|right|200px|A peppermint flower pot is shown on the terrace in New Belgrade, Serbia. Credit: [[c:user:VS6507|VS6507]].{{tlx|free media}}]]
Family: ''Lamiaceae''.

Peppermint (''Mentha'' × ''piperita'', also known as ''Mentha balsamea'' Wild)&lt;ref name= smsm&gt;{{cite book | title = Monographs on Selected Medicinal Plants | volume = Volume 2 |publisher= World Health Organization |location= Geneva |date=2002 |pages=188, 199 |isbn= 978-92-4-154537-2 |url= http://whqlibdoc.who.int/publications/2002/9241545372.pdf |accessdate=October 29, 2010}}&lt;/ref&gt; is a hybrid mint, a cross between ''Mentha aquatica'' (watermint) and ''Mentha spicata'' (spearmint).&lt;ref&gt;{{Cite book | title = The Complete Illustrated Book of Herbs | first = Alex | last = Frampton | publisher = The Reader's Digest Association | date = 2009 }}&lt;/ref&gt; Indigenous to Europe and the Middle East,&lt;ref&gt;{{cite web |title= Peppermint
|publisher= Botanical Online
|accessdate= 19 March 2018
|url= https://web.archive.org/web/20180319213600/https://www.botanical-online.com/english }}&lt;/ref&gt; the plant is now widely spread and cultivated in many regions of the world.&lt;ref name= empp&gt;{{ cite web
|title=Euro+Med Plantbase Project: ''Mentha'' × ''piperita'' 
|url=https://web.archive.org/web/20120309180913/http://ww2.bgbm.org/_EuroPlusMed/PTaxonDetail.asp?NameId=114331&amp;PTRefFk=500000 |date=9 March 2012 }}&lt;/ref&gt; It is occasionally found in the wild with its parent species.&lt;ref name= empp/&gt;&lt;ref name=fnwe&gt;{{ cite web
|title=Flora of NW Europe: ''Mentha'' × ''piperita''
|url= https://web.archive.org/web/20090919025128/http://ip30.eti.uva.nl/BIS/flora.php?selected=beschrijving&amp;menuentry=soorten&amp;id=3522 |date=September 19, 2009}}&lt;/ref&gt;

Peppermint essential oil has a high menthol content, also contains menthone and carboxyl esters, particularly menthyl acetate.&lt;ref&gt;{{cite book |title=PDR for Herbal Medicines |edition= 4th |author=Thomson Healthcare |page=640 |date=2007 |isbn=978-1-56363-678-3}}&lt;/ref&gt; Dried peppermint typically has 0.3–0.4% of volatile oil containing menthol (7–48%), menthone (20–46%), menthyl acetate (3–10%), menthofuran (1–17%) and 1,8-cineol (3–6%). Peppermint oil also contains small amounts of many additional compounds including limonene, pulegone, caryophyllene and pinene.&lt;ref&gt;{{cite book |title=Encyclopedia of Common Natural Ingredients used in food, drugs and cosmetics |url=https://archive.org/details/encyclopediaofco00leun |first=A. Y. |last=Leung |publisher=John Wiley &amp; Sons |location=New York |date=1980 |page=231 |isbn=9780471049548 }}&lt;/ref&gt;

Peppermint contains terpenoids and flavonoids such as eriocitrin, hesperidin, and kaempferol 7-O-rutinoside.&lt;ref&gt;{{cite journal | last1 = Dolzhenko | first1 = Yuliya | last2 = Bertea | first2 = Cinzia M. | last3 = Occhipinti | first3 = Andrea | last4 = Bossi | first4 = Simone | last5 = Maffei | first5 = Massimo E. | date = 2010 | title = UV-B modulates the interplay between terpenoids and flavonoids in peppermint (''Mentha'' × ''piperita L.'') | journal = Journal of Photochemistry and Photobiology B: Biology | volume = 100 | issue = 2| pages = 67–75 | doi = 10.1016/j.jphotobiol.2010.05.003 | pmid = 20627615 }}&lt;/ref&gt;

Peppermint oil has a high concentration of natural pesticides, mainly pulegone (found mainly in ''Mentha arvensis'' var. ''piperascens''  cornmint, field mint, Japanese mint, and to a lesser extent (6,530 ppm) in ''Mentha'' × ''piperita'' subsp. ''notho''&lt;ref&gt;Duke's Data Base http://www.ars-grin.gov/cgi-bin/duke/highchem.pl |date=December 2017 }}&lt;/ref&gt;) and menthone.&lt;ref name=&quot;Krieger2001&quot;&gt;{{cite book|author=Robert Irving Krieger|title=Handbook of Pesticide Toxicology: Principles|url=https://books.google.com/books?id=ib8Qhju9EQEC&amp;pg=PA823|accessdate=11 October 2010|date=2001|publisher=Academic Press|isbn=978-0-12-426260-7|page=823}}&lt;/ref&gt; It is known to repel some pest insects, including mosquitos, and has uses in organic gardening. It is also widely used to repel rodents.&lt;ref&gt;{{cite web|title= Peppermint Oil = rat repelent|date=21 May 2018|url=https://www.pests.org/how-to-get-rid-of-rats-naturally-peppermint-oil-black-pepper/}}&lt;/ref&gt;&lt;ref&gt;{{cite journal | title=Bioefficacy of Mentha piperita essential oil against dengue fever mosquito Aedes aegypti L |author1=Kumar, Sarita |author2=Wahab, Naim |author3=Warikoo, Radhika | date=April 2011 | journal=Asian Pacific Journal of Tropical Biomedicine | doi=10.1016/S2221-1691(11)60001-4 | pmid=23569733 | volume=1 | issue=2 | pages=85–8}}&lt;/ref&gt;&lt;ref&gt;{{cite book | title=Dear Dirt Doctor: Questions Answered the Natural Way | author=Garrett, Howard | date=2003 | publisher=University of Texas Press | url=https://books.google.com/books?id=k2rUBAAAQBAJ | page=54| isbn=9781477304143 }}&lt;/ref&gt;&lt;ref&gt;{{cite book | title=Industrial Crops and Uses | author=Singh, Bharat P. | date=2010 | publisher=Centre for Agriculture and Biosciences International | url=https://books.google.com/books?id=I1j2ZtzZXgUC | page=144| isbn=9781845936167 }}&lt;/ref&gt;

The chemical composition of the essential oil from peppermint (''Mentha'' × ''piperita'' L.) was analyzed by Flame ionization detector (GC/FID) and Gas chromatography–mass spectrometry (GC-MS}: menthol (40.7%) and menthone (23.4%), (±)-menthyl acetate, 1,8-cineole, limonene, beta-pinene, and beta-caryophyllene.&lt;ref&gt;{{cite journal | last1 = Schmidt | first1 = E. | last2 = Bail | first2 = S. | last3 = Buchbauer | first3 = G. | last4 = Stoilova | first4 = I. | last5 = Atanasova | first5 = T. | last6 = Stoyanova | first6 = A. | last7 = Krastanov | first7 = A. | last8 = Jirovetz | first8 = L. | date = 2009 | title = Chemical composition, olfactory evaluation and antioxidant effects of essential oil from Mentha x piperita | journal = Natural Product Communications | volume = 4 | issue = 8| pages = 1107–1112 | doi = 10.1177/1934578X0900400819 | pmid = 19768994 }}&lt;/ref&gt;

The Herbal Tea &quot;Cold &amp; Flu Time&quot; contains: Chinese honeysuckle, mulberry leaf, lophatherum, pueraria root, peppermint, licorice root, orange peel, from the box &quot;INGREDIENTS&quot;.
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==''Mitragyna speciosa''==
{{main|Remedy/Plants/Rubiaceae}}

==''Momordica charantia''==
[[Image:Momordica charantia dsc07812.jpg|thumb|right|250px|Bitter melon is ''Momordica charantia''. Credit: [[c:user:David Monniaux|David Monniaux]].{{tlx|free media}}]]
Family: ''Cucurbitaceae''.
{{clear}}

==''Morus alba''==

Family: ''Moraceae''.

Various extracts from ''Morus alba'' including kuwanon G, moracin M, steppogenin-4′-O-β-D-glucoside and mulberroside A have been suggested as having a variety of potentially-useful medical effects.&lt;ref&gt;[https://archive.today/20120909142249/http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6T8D-47MKSRX-7&amp;_user=10&amp;_coverDate=02/28/2003&amp;_rdoc=1&amp;_fmt=high&amp;_orig=search&amp;_origin=search&amp;_sort=d&amp;_docanchor=&amp;view=c&amp;_searchStrId=1450477094&amp;_rerunOrigin=google&amp;_acct=C000050221&amp;_version=1&amp;_urlVersion=0&amp;_userid=10&amp;md5=b0fe94317816caa28943642ed6522448&amp;searchtype=a &quot;Kuwanon G: an antibacterial agent from the root bark of ''Morus alba'' against oral pathogens&quot;]&lt;/ref&gt;&lt;ref&gt;Hypolipidemic and antioxidant effects of mulberry (''Morus alba'' L.) fruit in hyperlipidaemia rats Yang X., Yang L., Zheng H. Food and Chemical Toxicology 2010 48:8-9 (2374-2379)&lt;/ref&gt;&lt;ref&gt;Mulberry fruit protects dopaminergic neurons in toxin-induced Parkinson's disease models. Kim H.G., Ju M.S., Shim J.S., Kim M.C., Lee S.H., Huh Y., Kim S.Y., Oh M.S.  The British Journal of Nutrition 2010 104:1 (8-16)&lt;/ref&gt;&lt;ref&gt;Albanol a from the root bark of ''Morus alba'' L. induces apoptotic cell death in HL60 human leukemia cell line Kikuchi T., Nihei M., Nagai H., Fukushi H., Tabata K., Suzuki T., Akihisa T. Chemical and Pharmaceutical Bulletin 2010 58:4 (568-571)&lt;/ref&gt;&lt;ref&gt;In vivo hypoglycemic effects of phenolics from the root bark of ''Morus alba'' Zhang M., Chen M., Zhang H.-Q., Sun S., Xia B., Wu F.-H.  Fitoterapia 2009 80:8 (475-477)&lt;/ref&gt;&lt;ref&gt;Mulberroside A Possesses Potent Uricosuric and Nephroprotective Effects in Hyperuricemic Mice Wang C.-P., Wang Y., Wang X., Zhang X., Ye J.-F., Hu L.-S., Kong L.-D.  [Article in Press] Planta Medica 2010&lt;/ref&gt;&lt;ref&gt;{{Cite journal|pmid=20411402|year=2010|last1=Kim|first1=JK|last2=Kim|first2=M|last3=Cho|first3=SG|last4=Kim|first4=MK|last5=Kim|first5=SW|last6=Lim|first6=YH|title=Biotransformation of mulberroside a from Morus alba results in enhancement of tyrosinase inhibition|volume=37|issue=6|pages=631–7|doi=10.1007/s10295-010-0722-9|journal=Journal of Industrial Microbiology &amp; Biotechnology|s2cid=21236818}}&lt;/ref&gt;&lt;ref&gt;Adaptogenic effect of ''Morus alba'' on chronic footshock-induced stress in rats Nade V.S., Kawale L.A., Naik R.A., Yadav A.V.  Indian Journal of Pharmacology 2009 41:6&lt;/ref&gt;&lt;ref&gt;Mulberry leaf extract restores arterial pressure in streptozotocin-induced chronic diabetic rats Naowaboot J., Pannangpetch P., Kukongviriyapan V., Kukongviriyapan U., Nakmareong S., Itharat A.  Nutrition Research 2009 29:8 (602-608)&lt;/ref&gt;&lt;ref&gt;Antihyperglycemic, antioxidant and antiglycation activities of mulberry leaf extract in streptozotocin-induced chronic diabetic rats Naowaboot J., Pannangpetch P., Kukongviriyapan V., Kongyingyoes B., Kukongviriyapan U.  Plant Foods for Human Nutrition 2009 64:2 (116-121)&lt;/ref&gt;&lt;ref&gt;Neutralization of local and systemic toxicity of ''Daboia russelii'' venom by ''Morus alba'' plant leaf extract Chandrashekara K.T., Nagaraju S., Usha Nandini S., Basavaiah , Kemparaju K.  Phytotherapy Research 2009 23:8 (1082-1087)&lt;/ref&gt; Cyanidin-3-O-beta-ᴅ-glucopyranoside and Sanggenon G extracted from ''Morus alba'' were studied in animals models for some effects on the central nervous system, but clinical trials are necessary to confirm the effects.&lt;ref&gt;{{Cite journal|last1=Tam|first1=Dao Ngoc Hien|last2=Nam|first2=Nguyen Hai|last3=Elhady|first3=Mohamed Tamer|last4=Tran|first4=Linh|last5=Hassan|first5=Osama Gamal|last6=Sadik|first6=Mohamed|last7=Tien|first7=Phan Thi My|last8=Elshafei|first8=Ghada Amr|last9=Huy|first9=Nguyen Tien|date=2020-05-07|title=Effects of Mulberry on the Central Nervous System: A Literature Review|url=http://www.eurekaselect.com/181734/article|journal=Current Neuropharmacology|language=en|volume=18|issue=2|pages=193–219|doi=10.2174/1570159X18666200507081531|pmid=32379591|pmc=8033976}}&lt;/ref&gt;

''Morus alba'' is a traditional Chinese medicine that contains alkaloids and flavonoids that are bioactive compounds.&lt;ref&gt;{{Cite journal|last1=Zhang|first1=Hongxia|last2=Ma|first2=Zheng Feei|last3=Luo|first3=Xiaoqin|last4=Li|first4=Xinli|date=2018-05-21|title=Effects of Mulberry Fruit (Morus alba L.) Consumption on Health Outcomes: A Mini-Review|journal=Antioxidants|volume=7|issue=5|page=69|doi=10.3390/antiox7050069|issn=2076-3921|pmc=5981255|pmid=29883416|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2017-10-01|title=Phytopharmacological potential of different species of Morus alba and their bioactive phytochemicals: A review|journal=Asian Pacific Journal of Tropical Biomedicine|language=en|volume=7|issue=10|pages=950–956|doi=10.1016/j.apjtb.2017.09.015|issn=2221-1691|last1=Hussain|first1=Fahad|last2=Rana|first2=Zohaib|last3=Shafique|first3=Hassan|last4=Malik|first4=Arif|last5=Hussain|first5=Zahid|doi-access=free}}&lt;/ref&gt; Studies have shown that these compounds may help reduce high cholesterol, obesity, and stress.&lt;ref&gt;{{Cite journal|last1=Metwally|first1=Fateheya Mohamed|last2=Rashad|first2=Hend|last3=Mahmoud|first3=Asmaa Ahmed|date=March 2019|title=Morus alba L. Diminishes visceral adiposity, insulin resistance, behavioral alterations via regulation of gene expression of leptin, resistin and adiponectin in rats fed a high-cholesterol diet|url=http://dx.doi.org/10.1016/j.physbeh.2018.12.010|journal=Physiology &amp; Behavior|volume=201|pages=1–11|doi=10.1016/j.physbeh.2018.12.010|pmid=30552920|s2cid=54482222|issn=0031-9384}}&lt;/ref&gt;

==''Morus indica''==

''Morus indica'' is often grown for its medicinal properties. As with most berries, the mulberries of ''M. indica'' have potent antioxidant properties.&lt;ref name=&quot;:2&quot;&gt;{{Cite journal |last1=Andallu |first1=Bondada |last2=Varadacharyulu |first2=N. Ch |year=2003 |title=Antioxidant role of mulberry (Morus indica L. cv. Anantha) leaves in streptozotocin-diabetic rats |journal=Clinica Chimica Acta; International Journal of Clinical Chemistry |volume=338 |issue=1–2 |pages=3–10 |issn=0009-8981 |pmid=14637259|doi=10.1016/S0009-8981(03)00322-X }}&lt;/ref&gt; The primary medicinal use of ''Morus indica'' is as a method of regulating blood glucose levels in diabetic patients. Multiple studies in humans and mice have found that the use of ''Morus indica'' lowered the blood glucose levels of diabetics through multiple different pathways.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref&gt;{{Cite journal |last1=Andallu |first1=B. |last2=Suryakantham |first2=V. |last3=Lakshmi Srikanthi |first3=B. |last4=Reddy |first4=G. K. |year=2001 |title=Effect of mulberry (Morus indica L.) therapy on plasma and erythrocyte membrane lipids in patients with type 2 diabetes |journal=Clinica Chimica Acta; International Journal of Clinical Chemistry |volume=314 |issue=1–2 |pages=47–53 |issn=0009-8981 |pmid=11718678|doi=10.1016/S0009-8981(01)00632-5 }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal |url=https://www.sciencedirect.com/science/article/pii/S1674638412600458 |title=Effects of Flavonoids in Morus indica on Blood Lipids and Glucose in Hyperlipidemia-diabetic Rats |issue=4 |pages=314–318 |journal=Chinese Herbal Medicines |volume=4 |accessdate=5 April 2019|doi=10.3969/j.issn.1674-6348.2012.04.008 |date=November 2012 }}&lt;/ref&gt;

==''Morus mongolica''==

''Morus mongolica'' is known to have multiple flavonoid and phenolic compounds.&lt;ref&gt;{{Cite journal|last1=Sohn|first1=H. Y.|last2=Son|first2=K. H.|last3=Kwon|first3=C. S.|last4=Kwon|first4=G. S.|last5=Kang|first5=S. S.|date=November 2004|title=Antimicrobial and cytotoxic activity of 18 prenylated flavonoids isolated from medicinal plants: Morus alba L., Morus mongolica Schneider, Broussnetia papyrifera (L.) Vent, Sophora flavescens Ait and Echinosophora koreensis Nakai|journal=Phytomedicine|volume=11|issue=7–8|pages=666–672|doi=10.1016/j.phymed.2003.09.005|issn=0944-7113|pmid=15636183}}&lt;/ref&gt;&lt;ref name=Zhang2010&gt;{{Cite journal|last1=Zhang|first1=Xiao-Qi|last2=Jing|first2=Ying|last3=Wang|first3=Guo-Cai|last4=Wang|first4=Ying|last5=Zhao|first5=Hui-Nan|last6=Ye|first6=Wen-Cai|date=October 2010|title=Four new flavonoids from the leaves of Morus mongolica|journal=Fitoterapia|volume=81|issue=7|pages=813–815|doi=10.1016/j.fitote.2010.04.010|pmid=20450963|issn=0367-326X}}&lt;/ref&gt;&lt;ref name=Chen2017&gt;{{Cite journal|last1=Chen|first1=Hu|last2=Yu|first2=Wansha|last3=Chen|first3=Guo|last4=Meng|first4=Shuai|last5=Xiang|first5=Zhonghuai|last6=He|first6=Ningjia|date=December 21, 2017|title=Antinociceptive and Antibacterial Properties of Anthocyanins and Flavonols from Fruits of Black and Non-Black Mulberries|journal= Molecules|publication-date=January 2018|volume=23|issue=1|pages=4|doi=10.3390/molecules23010004|issn=1420-3049|pmc=5943937|pmid=29267231}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Huang|first1=Lian|last2=Fuchino|first2=Hiroyuki|last3=Kawahara|first3=Nobuo|last4=Narukawa|first4=Yuji|last5=Hada|first5=Noriyasu|last6=Kiuchi|first6=Fumiyuki|date=October 2016|title=Application of a new method, orthogonal projection to latent structure (OPLS) combined with principal component analysis (PCA), to screening of prostaglandin E2 production inhibitory flavonoids in Scutellaria Root|journal=Journal of Natural Medicines|volume=70|issue=4|pages=731–739|doi=10.1007/s11418-016-1004-2|pmid=27164908|s2cid=15105430|issn=1340-3443}}&lt;/ref&gt; These compounds can be found in the fruits,&lt;ref name=Chen2017/&gt; leaves,&lt;ref name=Zhang2010/&gt; and bark.&lt;ref&gt;{{Cite journal|last1=Shi|first1=Ya-Qin|last2=Fukai|first2=Toshio|last3=Sakagami|first3=Hiroshi|last4=Chang|first4=Wen-Jin|last5=Yang|first5=Pei-Quan|last6=Wang|first6=Feng-Peng|last7=Nomura|first7=Taro|date=February 2001|title=Cytotoxic Flavonoids with Isoprenoid Groups from Morus mongolica|journal=Journal of Natural Products|volume=64|issue=2|pages=181–188|doi=10.1021/np000317c|pmid=11429996|issn=0163-3864}}&lt;/ref&gt;

==''Morus nigra''==
[[Image:Shahtoot.JPG|thumb|right|200px|A full-grown shahtoot is shown. Credit: [[c:user:Ayda D|Ayda D]]{{tlx|free media}}]]
''Morus'', a genus of flowering plants in the family ''Moraceae'', consists of diverse species of deciduous trees commonly known as mulberries, growing wild and under cultivation in many temperate world regions.&lt;ref name=&quot;suttie&quot;&gt;{{cite web |author1=J.M. Suttie |title=''Morus alba'' L. |url=http://www.fao.org/ag/AGP/AGPC/doc/Gbase/data/pf000542.htm |publisher=United Nations, Food and Agriculture Organization |accessdate=8 March 2020 |date=2002}}&lt;/ref&gt;&lt;ref name=Morus/&gt;&lt;ref name=Duke1983/&gt; Generally, the genus has three well-known species ostensibly named for the fruit color of the best-known cultivar: white, red, and black mulberry (''Morus alba'', ''M. rubra'', and ''M. nigra'', respectively), with numerous cultivars,&lt;ref name=&quot;crfg&quot;&gt;{{cite web |title=Mulberry |url=https://www.crfg.org/pubs/ff/mulberry.html |publisher=California Rare Fruit Growers |access-date=8 March 2020 |date=1997}}&lt;/ref&gt;&lt;ref name=&quot;kew&quot;&gt;{{cite web |url=http://www.theplantlist.org/tpl1.1/search?q=Morus |publisher=The Plant List, Kew Botanic Gardens |title=Search for ''Morus'' |date=2013}}&lt;/ref&gt; The name &quot;white mulberry&quot; came about because the first specimens named by European taxonomists were a cultivated mutation prized for their white fruit, but wild trees bear black fruit like other mulberries. White mulberry is native to South Asia, but is widely distributed across Europe, Southern Africa, South America, and North America.&lt;ref name=Morus&gt;{{cite web |title=''Morus nigra'' (black mulberry) |url=https://www.cabi.org/isc/datasheet/34830 |publisher=CABI |accessdate=8 March 2020 |date=20 November 2019}}&lt;/ref&gt; ''Morus alba'' is regarded as an invasive species in Brazil and the United States.&lt;ref name=Morus/&gt;

The closely related genus ''Broussonetia'' is also commonly known as mulberry, notably the paper mulberry, ''Broussonetia papyrifera''.&lt;ref name=fna&gt;Wunderlin, Richard P. (1997). &quot;Broussonetia papyrifera&quot;. In ''Flora of North America'' Editorial Committee (ed.). ''Flora of North America North of Mexico'' (FNA). 3. New York and Oxford – via eFloras.org, Missouri Botanical Garden, St. Louis, MO &amp; Harvard University Herbaria, Cambridge, MA.&lt;/ref&gt;

The fruit and leaves are sold in various forms as dietary supplements. Unripe fruit and green parts of the plant have a white sap that may be toxic, stimulating, or mildly hallucinogenic.&lt;ref&gt;{{cite web |title=White mulberry – ''Morus alba'' In: ''Ohio Perennial and Biennial Weed Guide'' |publisher=The Ohio State University |accessdate=20 October 2012 |url=https://web.archive.org/web/20120412062338/http://www.oardc.ohio-state.edu/weedguide/singlerecord.asp?id=200 |archive-date=2012-04-12}}&lt;/ref&gt;

Mulberry fruit color derives from anthocyanins,&lt;ref name=Duke1983/&gt; which have unknown effects in humans.&lt;ref name=&quot;efsa2010&quot;&gt;{{cite journal |title=Scientific opinion on the substantiation of health claims related to various food(s)/food constituent(s) and protection of cells from premature aging, antioxidant activity, antioxidant content and antioxidant properties, and protection of DNA, proteins and lipids from oxidative damage pursuant to Article 13(1) of Regulation (EC) No 1924/20061 |publisher=EFSA Panel on Dietetic Products, Nutrition and Allergies |journal=EFSA Journal |year=2010 |volume=8 |issue=2 |page=1489 |doi=10.2903/j.efsa.2010.1752|doi-access=free}}&lt;/ref&gt; Anthocyanins are responsible for the attractive colors of fresh plant foods, including orange, red, purple, black, and blue.&lt;ref name=efsa2010/&gt; These colors are water-soluble and easily extractable, yielding natural food colorants.&lt;ref name=Morus/&gt; Due to a growing demand for natural food colorants, they have numerous applications in the food industry.&lt;ref name=Duke1983/&gt;&lt;ref name=efsa2010/&gt;

''Morus nigra'', called black mulberry&lt;ref&gt;&quot;Morus nigra&quot;. Germplasm Resources Information Network (GRIN). Agricultural Research Service (ARS), United States Department of Agriculture (USDA). Retrieved 21 December 2017.&lt;/ref&gt; or blackberry (not to be confused with the blackberries that are various species of ''Rubus''),&lt;ref&gt;{{cite web |url=http://www.fao.org/waicent/faoinfo/economic/faodef/fdef08e.htm#8.06 |title=Definition And Classification Of Commodities (Draft) 8. Fruits And Derived Products |publisher=Food and Agriculture Organization of the United Nations |access-date=1 August 2014}}&lt;/ref&gt; is a species of flowering plant that is native to southwestern Asia and the Iberian Peninsula, where it has been cultivated for so long that its precise natural range is unknown.&lt;ref name=RHSAZMorus&gt;{{cite book|title=RHS A-Z encyclopedia of garden plants|year=2008|publisher=Dorling Kindersley|location=United Kingdom|isbn=978-1405332965|pages=1136}}&lt;/ref&gt; The black mulberry is known for its large number of chromosomes, 308 (44x ploidy).&lt;ref&gt;{{cite journal| last1=Zeng|first1=Q |last2=Chen |first2=H| title=Definition of Eight Mulberry Species in the Genus Morus by Internal Transcribed Spacer-Based Phylogeny. | journal=PLOS ONE| date=2015| volume=10| issue=8|pages=e0135411 |doi=10.1371/journal.pone.0135411 |pmid=26266951 |bibcode=2015PLoSO..1035411Z }}&lt;/ref&gt;
{{clear}}

==''Mucuna pruriens''==
{{main|Remedy/Plants/Fabaceae}}

==''Muira puama''==

Family: ''Olacaceae'', genus ''Ptychopetalum''.

==''Myristica argentea''==

==''Myristica fragrans''==
[[Image:Biji Pala Bubuk.jpg|thumb|right|200px|Seed are on the left or ground spice on the right. Credit: [[c:user:Herusutimbul|Herusutimbul]].{{tlx|free media}}]]
[[Image:Myristica Fragrans - ജാതിമരം.JPG|thumb|left|upright|200px|Nutmeg tree (''Myristica fragrans'') is shown. Credit: [[c:user:കാക്കര|കാക്കര]].{{tlx|free media}}]]
Family: ''Myristicaceae''

Nutmeg is the seed or ground spice of several species of the genus ''Myristica''.&lt;ref name=&quot;fao1994&quot;&gt;{{cite web
|title=Nutmeg and derivatives (Review)|date=September 1994|publisher=Food and Agriculture Organization (FAO) of the United Nations
|url=https://web.archive.org/web/20181030035405/http://www.fao.org/docrep/v4084e/v4084e00.htm#Contents|archive-date=30 October 2018|accessdate=29 October 2018}}&lt;/ref&gt; ''Myristica fragrans'' (fragrant nutmeg or true nutmeg) is a dark-leaved evergreen tree cultivated for two spices derived from its fruit: nutmeg, from its seed, and mace, from the seed covering. It is also a commercial source of an essential oil and nutmeg butter. Indonesia is the main producer of nutmeg and mace.

If consumed in amounts exceeding its typical use as a spice, nutmeg powder may produce allergic reactions, cause contact dermatitis, or have psychoactive effects.&lt;ref name=Nutmeg&gt;{{ cite web|url=https://www.drugs.com/npp/nutmeg.html|title=Nutmeg|publisher=Drugs.com|date=2009|accessdate=2017-05-04 }}&lt;/ref&gt; Although used in traditional medicine for treating various disorders, nutmeg has no scientifically confirmed prescription drug (medicinal value).&lt;ref name=Nutmeg/&gt;

Nutmeg is the spice made by grinding the seed of the fragrant nutmeg (''Myristica fragrans'') tree into powder.  The spice has a distinctive pungent fragrance and a warm, slightly sweet taste; it is used to flavor many kinds of baked goods, confections, puddings, potatoes, meats, sausages, sauces, vegetables, and such beverages as eggnog.&lt;ref name=&quot;ebo&quot;&gt;{{ cite encyclopedia | url = http://www.britannica.com/EBchecked/topic/422816/nutmeg | title = Nutmeg spice, In: ''Encyclopædia Britannica Online'' }}&lt;/ref&gt;

The seeds are dried gradually in the sun over a period of six to eight weeks. During this time the nutmeg shrinks away from its hard seed coat until the kernels rattle in their shells when shaken. The shell is then broken with a wooden club and the nutmegs are picked out. Dried nutmegs are grayish brown ovals with furrowed surfaces.&lt;ref name=ebo/&gt; The nutmegs are roughly egg-shaped, about {{Convert|20.5-30|mm|in|abbr=on}} long and {{Convert|15-18|mm|in|abbr=on}} wide, weighing {{Convert|5-10|g|oz|abbr=on}} dried.

The essential oil obtained by steam distillation of ground nutmeg&lt;ref name=&quot;fao&quot;&gt;{{cite web|title=Description of components of nutmeg |url=http://www.fao.org/docrep/v4084e/v4084e04.htm |publisher=Food and Agriculture Organization of the United Nations|date=September 1994|access-date=2017-04-13}}&lt;/ref&gt; is used in the perfumery and pharmaceutical industries. The volatile fraction contains dozens of terpenes and phenylpropanoids, including D-pinene, limonene, D-borneol, L-terpineol, geraniol, safrol, and myristicin.&lt;ref name=fao/&gt;&lt;ref&gt;{{cite journal|pmc=5222521|year=2016|last1=Abourashed|first1=E. A.|title=Chemical diversity and pharmacological significance of the secondary metabolites of nutmeg (''Myristica fragrans'' Houtt.)|journal=Phytochemistry Reviews|volume=15|issue=6|pages=1035–1056|last2=El-Alfy|first2=A. T.|pmid=28082856|doi=10.1007/s11101-016-9469-x}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|pmid=22429024|year=2012|last1=Piras|first1=A.|title=Extraction and separation of volatile and fixed oils from seeds of ''Myristica fragrans'' by supercritical CO&lt;sub&gt;2&lt;/sub&gt;: Chemical composition and cytotoxic activity on Caco-2 cancer cells|journal=Journal of Food Science|volume=77|issue=4|pages=C448–53|last2=Rosa|first2=A.|last3=Marongiu|first3=B.|last4=Atzeri|first4=A.|last5=Dessì|first5=M. A.|last6=Falconieri|first6=D.|last7=Porcedda|first7=S.|doi=10.1111/j.1750-3841.2012.02618.x}}&lt;/ref&gt; In its pure form, myristicin is a toxin, and consumption of excessive amounts of nutmeg can result in myristicin poisoning.&lt;ref name=&quot;jmt&quot;&gt;{{cite journal|pmc=4057546|date=2014|last1=Ehrenpreis|first1=J. E.|title=Nutmeg Poisonings: A Retrospective Review of 10&amp;nbsp;Years Experience from the Illinois Poison Center, 2001–2011|journal=Journal of Medical Toxicology|volume=10|issue=2|pages=148–151|last2=Deslauriers|first2=C|last3=Lank|first3=P|last4=Armstrong|first4=P. K.|last5=Leikin|first5=J. B.|doi=10.1007/s13181-013-0379-7|pmid=24452991}}&lt;/ref&gt;

The oil is colorless or light yellow, and smells and tastes of nutmeg. It is used as a natural food flavoring in baked goods, syrups, beverages, and sweets. It is used to replace ground nutmeg, as it leaves no particles in the food. The essential oil is also used in the manufacturing of toothpaste and cough syrups.&lt;ref&gt;{{Cite book|url=https://books.google.com/books?id=dKA5DwAAQBAJ&amp;q=nutmeg+essential+oil+toothpaste+cough+syrup&amp;pg=PA21|title=Grenada: Carriacou and Petite Martinique|last=Crask|first=Paul|date=2017-11-05|publisher=Bradt Travel Guides|isbn=9781784770624 }}&lt;/ref&gt;

Nutmeg butter is obtained from the nut by expression, is semisolid, reddish-brown in colour, and has the taste and smell of nutmeg itself.&lt;ref name=fao/&gt; About 75% (by weight) of nutmeg butter is trimyristin, which can be turned into myristic acid, a 14-carbon fatty acid, which can be used as a replacement for cocoa butter, can be mixed with other fats like cottonseed oil or palm oil, and has applications as an industrial lubricant.

Two other species of genus ''Myristica'' with different flavors, ''Myristica malabarica'' and ''Myristica argentea'', are sometimes used to adulterate nutmeg as a spice.&lt;ref name=clovegarden&gt;{{cite web |url=http://www.clovegarden.com/ingred/sp_nutmegz.html |title=Nutmeg |website=www.clovegarden.com |accessdate=2017-07-22}}&lt;/ref&gt;
{{clear}}

==''Myristica malabarica''==

==''Oryza barthii''==

Family: ''Poaceae''.

==''Oryza glaberrima''==

Family: ''Poaceae''.

==''Oryza rufipogon''==

Family: ''Poaceae''.

==''Oryza sativa''==

Family: ''Poaceae''.

==''Osmanthus fragrans''==

Family: ''Oleaceae''.

''Osmanthus fragrans'', variously known as sweet osmanthus, sweet olive, tea olive, and fragrant olive, is a species native to Asia from the Himalayas through southern China (Guizhou, Sichuan and Yunnan) to Taiwan, southern Japan and Southeast Asia as far south as Cambodia and Thailand.&lt;ref name=foc&gt;Flora of China: [http://www.efloras.org/florataxon.aspx?flora_id=2&amp;taxon_id=210001392 ''Osmanthus fragrans'']&lt;/ref&gt;&lt;ref name=fop&gt;Flora of Pakistan: [http://www.efloras.org/florataxon.aspx?flora_id=5&amp;taxon_id=210001392 ''Osmanthus fragrans'']&lt;/ref&gt;&lt;ref&gt;[http://apps.kew.org/wcsp/namedetail.do?name_id=354988 Kew World Checklist of Selected Plant Families, ''Osmanthus fragrans'']&lt;/ref&gt;&lt;ref&gt;[https://www.biodiversitylibrary.org/page/653228#page/47/mode/1up Loureiro, João de. 1790. ora Cochinchinensis 1: 29, ''Osmanthus fragrans'']&lt;/ref&gt;

Osmanthus tea has been used as an herbal tea for the treatment of irregular menstruation.&lt;ref&gt;Zhou S.,&quot;Flower herbal tea used for treatment of menopathies&quot;.,  ''Journal of Traditional Chinese Medicine'' 2008 28:3 (202–204)&lt;/ref&gt; The extract of dried flowers showed neuroprotective, free-radical scavenging, antioxidative effects in ''in vitro'' assays.&lt;ref&gt;Lee H.-H., Lin C.-T., Yang L.-L. &quot;Neuroprotection and free radical scavenging effects of Osmanthus fragrans.&quot;,  Journal of Biomedical Science 2007 14:6 (819–827)&lt;/ref&gt;

==''Panax ginseng''==

Family: ''Araliaceae''.

==''Persea americana''==

Family:	''Lauraceae''.

Avocados have diverse fats.&lt;ref name=ND&gt;{{cite web|url = http://nutritiondata.self.com/facts/fruits-and-fruit-juices/1843/2|title = Avocados, raw, all commercial varieties, per 100 grams|publisher = NutritionData.com|accessdate = 17 April 2013|date = 2013}}&lt;/ref&gt; For a typical one:
* About 75% of an avocado's energy comes from fat, most of which (67% of total fat) is monounsaturated fat as oleic acid.&lt;ref name=ND/&gt;
* Other predominant fats include palmitic acid and linoleic acid.&lt;ref name=ND/&gt;
* The saturated fat content amounts to 14% of the total fat.&lt;ref name=ND/&gt; 
* Typical total fat composition is roughly: 1%&amp;nbsp;omega-3 fatty acid (ω-3), 14%&amp;nbsp;omega-6 fatty acid (ω-6), 71%&amp;nbsp;omega-9 fatty acid (ω-9) (65% oleic and 6% palmitoleic), and 14% saturated fat (palmitic acid).&lt;ref name=ND/&gt;

==''Phellodendron amurense''==

Family: ''Rutaceae''.

Berberine is found in ''Phellodendron amurense'' (Amur cork tree).&lt;ref&gt;{{cite journal | author = Zhang Q, Cai L, Zhong G, Luo W | title = Simultaneous determination of jatrorrhizine, palmatine, berberine, and obacunone in Phellodendri Amurensis Cortex by RP-HPLC | journal = Zhongguo Zhong Yao Za Zhi = Zhongguo Zhongyao Zazhi = China Journal of Chinese Materia Medica | volume = 35 | issue = 16 | pages = 2061–4 | date = 2010 | pmid = 21046728 | doi = 10.4268/cjcmm20101603 }}&lt;/ref&gt;

==''Pistacia vera''==
[[Image:860631-Pistachio-IMG 6862-2.jpg|thumb|right|250px|Torbat-e-Heydariyeh pistachio farms are in Razavi Khorasan Province, Iran - September 22, 2007. Credit: [[c:user:Safa.daneshvar|Safa.daneshvar]].{{tlx|free media}}]]
Family ''Anacardiaceae''.

Raw pistachios are 4% water, 45% fat, 28% carbohydrates, and 20% protein. In a 100-gram reference amount, pistachios provide {{convert|2351|kJ|kcal}} of food energy and are a rich source (20% or more of the Daily Value or DV) of protein, dietary fiber, several dietary minerals, and the B vitamins, thiamin (76% DV) and vitamin B&lt;sub&gt;6&lt;/sub&gt; (131% DV) (table).&lt;ref name=usda/&gt; Pistachios are a moderate source (10–19% DV) of calcium, riboflavin, pantothenic acid (vitamin B&lt;sub&gt;5&lt;/sub&gt;), folate, vitamin E, and vitamin K.

The fat profile of raw pistachios consists of saturated fats, monounsaturated fats and polyunsaturated fats.&lt;ref name=&quot;usda&quot;&gt;{{cite web | url=https://ndb.nal.usda.gov/ndb/foods/show/3687?fgcd=&amp;man=&amp;lfacet=&amp;count=&amp;max=&amp;sort=&amp;qlookup=&amp;offset=&amp;format=Full&amp;new=&amp;measureby= | title=Pistachio nuts, raw per 100 g | publisher=USDA National Nutrient Database | work=Release SR-28 | date=2016 | accessdate=20 May 2016}}&lt;/ref&gt;&lt;ref name=okay/&gt; Saturated fatty acids include palmitic acid (10% of total) and stearic acid (2%).&lt;ref name=okay/&gt; Oleic acid is the most common monounsaturated fatty acid (51% of total fat)&lt;ref name=&quot;okay&quot;&gt;{{cite journal|journal=Die Gartenbauwissenschaft|volume=67|issue=3|year=2002|title=The comparison of some pistachio cultivars regarding their fat, fatty acids and protein content|author=Okay Y|jstor=24137567|pages=107–113 }}&lt;/ref&gt; and linoleic acid, a polyunsaturated fatty acid, is 31% of total fat.&lt;ref name=usda/&gt; Relative to other tree nuts, pistachios have a lower amount of fat and food energy but higher amounts of potassium, vitamin K, Tocopherol (γ-tocopherol), and certain phytochemicals such as carotenoids, and phytosterols.&lt;ref name=&quot;Bulló2015&quot;&gt;{{cite journal|last1=Bulló|first1=M|last2=Juanola-Falgarona|first2=M|last3=Hernández-Alonso|first3=P|last4=Salas-Salvadó|first4=J|title=Nutrition attributes and health effects of pistachio nuts|journal=The British Journal of Nutrition|date=April 2015|volume=113|issue=Supplement 2|pages=S79-93|doi=10.1017/S0007114514003250|pmid=26148925|type=Review }}&lt;/ref&gt;&lt;ref name=&quot;Dreher2012&quot;&gt;{{cite journal|last1=Dreher|first1=ML|title=Pistachio nuts: composition and potential health benefits|journal=Nutrition Reviews|date=April 2012|volume=70|issue=4|pages=234–40|doi=10.1111/j.1753-4887.2011.00467.x|pmid=22458696|type=Review}}&lt;/ref&gt;
{{clear}}

==''Prunus avium''==
{{main|Remedy/Plants/Rosaceae}}

==''Prunus cerasus''==
{{main|Remedy/Plants/Rosaceae}}

==''Pueraria mirifica''==
{{main|Remedy/Plants/Fabaceae}}

==''Pueraria tuberosa''==
{{main|Remedy/Plants/Fabaceae}}

==''Pyropia plicata''==

Family: ''Bangiaceae''

==''Pyropia tenera''==

Family: ''Bangiaceae''

==''Pyropia tenuipedalis''==

Family: ''Bangiaceae''

==''Porphyra umbilicalis''==

Family: ''Bangiaceae''

==''Pyropia yezoensis''==

Family: ''Bangiaceae''

&quot;A unique life cycle transition in the red seaweed ''Pyropia yezoensis'' depends on apospory.&quot;&lt;ref name=Mikami&gt;{{ cite journal
|author=Koji Mikami, Chengze Li, Ryunosuke Irie &amp; Yoichiro Hama
|title=A unique life cycle transition in the red seaweed ''Pyropia yezoensis'' depends on apospory
|journal=Nature Communications Biology
|date=7 August 2019
|volume=2
|issue=
|pages=299
|url=https://www.nature.com/articles/s42003-019-0549-5
|arxiv=
|bibcode=
|doi=10.1038/s42003-019-0549-5
|pmid=
|accessdate=15 April 2022 }}&lt;/ref&gt;

==''Rehmannia glutinosa''==

A number of chemical constituents including iridoids, phenethyl alcohol, glycosides, cyclopentanoid monoterpenes, and norcarotenoids, have been reported from the fresh or processed roots of ''Rehmannia glutinosa''.&lt;ref name=Oh&gt;{{cite journal | doi = 10.1002/chin.200602189 | title = Remophilanetriol: A New Eremophilane from the Roots of Rehmannia glutinosa | year = 2006 | last1 = Oh | first1 = Hyuncheol | journal = ChemInform | volume = 37 | issue = 2}}&lt;/ref&gt;

==''Rosa canina''==
{{main|Remedy/Plants/Rosaceae}}

==''Rubus allegheniensis''==
{{main|Remedy/Plants/Rosaceae}}

==''Rubus pensilvanicus''==
{{main|Remedy/Plants/Rosaceae}}

==''Rubus plicatus''==
{{main|Remedy/Plants/Rosaceae}}

==''Rubus vestitus''==
{{main|Remedy/Plants/Rosaceae}}

==''Sambucus callicarpa''==
{{main|Remedy/Plants/Adoxaceae}}

==''Sambucus mexicana''==
{{main|Remedy/Plants/Adoxaceae}}

==''Sambucus nigra''==
{{main|Remedy/Plants/Adoxaceae}}

==''Sambucus palmensis''==
{{main|Remedy/Plants/Adoxaceae}}

==''Sambucus peruviana''==
{{main|Remedy/Plants/Adoxaceae}}

==''Scutellaria baicalensis''==
[[Image:Scutellaria baicalensis flowers.jpg|thumb|right|250px|''Scutellaria baicalensis'' flowers and leaves are shown. Credit: [[c:user:Doronenko|Doronenko]].{{tlx|free media}}]]
Family: ''Lamiaceae''.

The main compounds responsible for the biological activity of skullcap are flavonoids.&lt;ref name=Guo&gt;{{cite journal |last1=Guo |first1=Xiaorong |last2=Wang |first2=Xiaoguo |last3=Su |first3=Wenhua |last4=Zhang |first4=Guangfei |last5=Zhou |first5=Rui |year=2011 |title=DNA Barcodes for Discriminating the Medicinal Plant ''Scutellaria baicalensis'' (''Lamiaceae'') and Its Adulterants |journal=Biological &amp; Pharmaceutical Bulletin |volume=34 |issue=8 |pages=1198–203 |pmid=21804206 |doi=10.1248/bpb.34.1198 }}&lt;/ref&gt; Baicalein, one of the important ''Scutellaria'' flavonoids, was shown to have cardiovascular effects in ''in vitro''.&lt;ref&gt;{{cite journal |last1=Huang |first1=Yu |last2=Tsang |first2=Suk-Ying |last3=Yao |first3=Xiaoqiang |last4=Chen|first4=Zhen-Yu |year=2005 |title=Biological Properties of Baicalein in Cardiovascular System |journal=Current Drug Targets |volume=5 |issue=2 |pages=177–84 |pmid=15853750 |doi=10.2174/1568006043586206}}&lt;/ref&gt; Research also shows that ''Scutellaria'' root modulates inflammatory activity ''in vitro'' to inhibit nitric oxide (NO), cytokine, chemokine and growth factor production in macrophages.&lt;ref&gt;{{cite journal |last1=Kim |first1=Eun Hye |last2=Shim |first2=Bumsang |last3=Kang |first3=Seunghee |last4=Jeong |first4=Gajin |last5=Lee |first5=Jong-soo |last6=Yu |first6=Young-Beob |last7=Chun |first7=Mison |year=2009 |title=Anti-inflammatory effects of ''Scutellaria baicalensis'' extract via suppression of immune modulators and MAP kinase signaling molecules |journal=Journal of Ethnopharmacology |volume=126 |issue=2 |pages=320–31 |pmid=19699788 |doi=10.1016/j.jep.2009.08.027}}&lt;/ref&gt; Isolated chemical compounds including wogonin, wogonoside, and 3,5,7,2',6'-pentahydroxyl flavanone found in ''Scutellaria'' have been shown to inhibit histamine and leukotriene release.&lt;ref&gt;{{cite journal |last1=Lim |first1=Beong Ou |year=2003 |title=Effects of wogonin, wogonoside, and 3,5,7,2′,6′-pentahydroxyflavone on chemical mediator production in peritoneal exduate cells and immunoglobulin E of rat mesenteric lymph node lymphocytes |journal=Journal of Ethnopharmacology |volume=84 |issue=1 |pages=23–9 |pmid=12499072 |doi=10.1016/S0378-8741(02)00257-X}}&lt;/ref&gt; Other active constituents include baicalin, apigenin, oroxylin A, scutellarein, and skullcapflavone.&lt;ref name=&quot;pmid14692724&quot;&gt;{{cite journal | authors = Awad R, Arnason JT, Trudeau V, Bergeron C, Budzinski JW, Foster BC, Merali Z | title = Phytochemical and biological analysis of skullcap (''Scutellaria lateriflora'' L.): a medicinal plant with anxiolytic properties | journal = Phytomedicine | volume = 10 | issue = 8 | pages = 640–9 | date = 2003 | pmid = 14692724 | doi = 10.1078/0944-7113-00374 }}&lt;/ref&gt;

A variety of flavonoids in ''Scutellaria'' species have been found to bind to the benzodiazepine site and/or a non-benzodiazepine site of the GABA&lt;sub&gt;A&lt;/sub&gt; receptor, including baicalin, baicalein, wogonin, apigenin, oroxylin A, scutellarein, and skullcapflavone II.&lt;ref name=&quot;pmid12494329&quot;&gt;{{cite journal | authors = Wang H, Hui KM, Chen Y, Xu S, Wong JT, Xue H | title = Structure-activity relationships of flavonoids, isolated from ''Scutellaria baicalensis'', binding to benzodiazepine site of GABA(A) receptor complex | journal = Planta Med. | volume = 68 | issue = 12 | pages = 1059–62 | year = 2002 | pmid = 12494329 | doi = 10.1055/s-2002-36357 }}&lt;/ref&gt;&lt;ref name=&quot;pmid10705749&quot;&gt;{{cite journal | authors = Hui KM, Wang XH, Xue H | title = Interaction of flavones from the roots of Scutellaria baicalensis with the benzodiazepine site | journal = Planta Med. | volume = 66 | issue = 1 | pages = 91–3 | year = 2000 | pmid = 10705749 | doi = 10.1055/s-0029-1243121 }}&lt;/ref&gt;&lt;ref name=&quot;pmid9776664&quot;&gt;{{cite journal | authors = Liao JF, Wang HH, Chen MC, Chen CC, Chen CF | title = Benzodiazepine binding site-interactive flavones from Scutellaria baicalensis root | journal = Planta Medica | volume = 64 | issue = 6 | pages = 571–2 | date = 1998 | pmid = 9776664 | doi = 10.1055/s-2006-957517 }}&lt;/ref&gt; Baicalin and baicalein,&lt;ref name=&quot;pmid9776664&quot;/&gt;&lt;ref name=&quot;CooperYamaguchi2004&quot;&gt;{{cite book|author1=Edwin Lowell Cooper|author2=Nobuo Yamaguchi|title=Complementary and Alternative Approaches to Biomedicine|url=https://archive.org/details/springer_10.1007-978-1-4757-4820-8|date=1 January 2004|publisher=Springer Science &amp; Business Media|isbn=978-0-306-48288-5|pages=[https://archive.org/details/springer_10.1007-978-1-4757-4820-8/page/n201 188]–}}&lt;/ref&gt;&lt;ref name=&quot;pmid18723037&quot;&gt;{{cite journal | authors = Wang F, Xu Z, Ren L, Tsang SY, Xue H | title = GABA A receptor subtype selectivity underlying selective anxiolytic effect of baicalin | journal = Neuropharmacology | volume = 55 | issue = 7 | pages = 1231–7 | date = 2008 | pmid = 18723037 | doi = 10.1016/j.neuropharm.2008.07.040 | s2cid = 20133964 }}&lt;/ref&gt;&lt;ref name=&quot;pmid12620506&quot;&gt;{{cite journal | vauthors = Liao JF, Hung WY, Chen CF | title = Anxiolytic-like effects of baicalein and baicalin in the Vogel conflict test in mice | journal = Eur. J. Pharmacol. | volume = 464 | issue = 2–3 | pages = 141–6 | date = 2003 | pmid = 12620506 | doi = 10.1016/s0014-2999(03)01422-5}}&lt;/ref&gt; wogonin,&lt;ref name=&quot;pmid12392823&quot;&gt;{{cite journal | authors = Hui KM, Huen MS, Wang HY, Zheng H, Sigel E, Baur R, Ren H, Li ZW, Wong JT, Xue H | title = Anxiolytic effect of wogonin, a benzodiazepine receptor ligand isolated from Scutellaria baicalensis Georgi | journal = Biochem. Pharmacol. | volume = 64 | issue = 9 | pages = 1415–24 | year = 2002 | pmid = 12392823 | doi = 10.1016/s0006-2952(02)01347-3}}&lt;/ref&gt; and apigenin&lt;ref name=&quot;pmid7617761&quot;&gt;{{cite journal | authors = Viola H, Wasowski C, Levi de Stein M, Wolfman C, Silveira R, Dajas F, Medina JH, Paladini AC | title = Apigenin, a component of Matricaria recutita flowers, is a central benzodiazepine receptors-ligand with anxiolytic effects | journal = Planta Medica | volume = 61 | issue = 3 | pages = 213–6 | date = 1995 | pmid = 7617761 | doi = 10.1055/s-2006-958058 }}&lt;/ref&gt; have been confirmed to act as positive allosteric modulators and produce anxiolytic effects in animals, whereas oroxylin A acts as a negative allosteric modulator (and also, notably, as a dopamine reuptake inhibitor).&lt;ref name=&quot;pmid12818372&quot;&gt;{{cite journal | authors = Huen MS, Leung JW, Ng W, Lui WS, Chan MN, Wong JT, Xue H | title = 5,7-Dihydroxy-6-methoxyflavone, a benzodiazepine site ligand isolated from Scutellaria baicalensis Georgi, with selective antagonistic properties | journal = Biochem. Pharmacol. | volume = 66 | issue = 1 | pages = 125–32 | year = 2003 | pmid = 12818372 | doi = 10.1016/s0006-2952(03)00233-8}}&lt;/ref&gt;&lt;ref name=&quot;pmid23543630&quot;&gt;{{cite journal |authors = Liu X, Hong SI, Park SJ, Dela Peña JB, Che H, Yoon SY, Kim DH, Kim JM, Cai M, Risbrough V, Geyer MA, Shin CY, Cheong JH, Park H, Lew JH, Ryu JH | title = The ameliorating effects of 5,7-dihydroxy-6-methoxy-2(4-phenoxyphenyl)-4H-chromene-4-one, an oroxylin A derivative, against memory impairment and sensorimotor gating deficit in mice | journal = Arch. Pharm. Res. | volume = 36 | issue = 7 | pages = 854–63 | date = 2013 | pmid = 23543630 | doi = 10.1007/s12272-013-0106-6 }}&lt;/ref&gt;&lt;ref name=&quot;Yoondela Peña2013&quot;&gt;{{cite journal|last1=Yoon|first1=Seo Young|last2=dela Peña|first2=Ike|last3=Kim|first3=Sung Mok|last4=Woo|first4=Tae Sun|last5=Shin|first5=Chan Young|last6=Son|first6=Kun Ho|last7=Park|first7=Haeil|last8=Lee|first8=Yong Soo|last9=Ryu|first9=Jong Hoon|last10=Jin|first10=Mingli|last11=Kim|first11=Kyeong-Man|last12=Cheong|first12=Jae Hoon|title=Oroxylin A improves attention deficit hyperactivity disorder-like behaviors in the spontaneously hypertensive rat and inhibits reuptake of dopamine in vitro|journal=Archives of Pharmacal Research|volume=36|issue=1|date=2013|pages=134–140|issn=0253-6269|doi=10.1007/s12272-013-0009-6|pmid=23371806|s2cid=23927252}}&lt;/ref&gt; As such, these compounds and actions, save oroxylin A, are likely to underlie the anxiolytic effects of ''Scutellaria'' species.&lt;ref name=&quot;Schwartz2008&quot;&gt;{{cite book|author=Stefanie Schwartz|title=Psychoactive Herbs in Veterinary Behavior Medicine|url=https://books.google.com/books?id=ZP6QVep-x24C&amp;pg=PA139|date=9 January 2008|publisher=John Wiley &amp; Sons|isbn=978-0-470-34434-7|pages=139–}}&lt;/ref&gt;

Scutellaria also contains rosmarinic acid which inhibits GABA transaminase which breaks GABA down, thus making it available longer.&lt;ref&gt;{{Cite journal|doi = 10.1177/1934578X1400900923|title = Molecular Cloning and Characterization of Tyrosine Aminotransferase and Hydroxyphenylpyruvate Reductase, and Rosmarinic Acid Accumulation in Scutellaria baicalensis|date = 2014|last1 = Kim|first1 = Yeon Bok|last2 = Uddin|first2 = Md Romij|last3 = Kim|first3 = Yeji|last4 = Park|first4 = Chun Geon|last5 = Park|first5 = Sang Un|journal = Natural Product Communications|volume = 9|issue = 9|pages = 1311–4|pmid = 25918800 }}&lt;/ref&gt;
{{clear}}

==''Senna alexandrina''==
{{main|Remedy/Plants/Fabaceae}}

==''Sideritis leucantha''==

Family: ''Lamiaceae''.

Hypolaetin 8-glucoside can be found in ''Sideritis leucantha''.&lt;ref name=Tomas&gt;Hypolaetin 8-glucoside from Sideritis leucantha. Francisco Tomas, Bernard Voirina, Francisco A.T. Barberan and Philippe Lebreton, Phytochemistry, 1985, Volume 24, Issue 7, Pages 1617–1618, doi:10.1016/S0031-9422(00)81082-1&lt;/ref&gt;

==''Silybum marianum''==
{{main|Remedy/Plants/Asteraceae}}

==''Simarouba glauca''==
[[Image:Simarouba glauca Flower.jpg|thumb|right|250px|Flowers of the paradise tree are captured during flowering season. Credit: [[c:user:GunasekarVV|GunasekarVV]].{{tlx|free media}}]]
Family:	''Simaroubaceae''

Though there is some research&lt;ref&gt;{{Cite web|url=http://www.rain-tree.com/simaruba.htm|title=Simarouba (Simarouba glauca) Database file in the Tropical Plant Database of herbal remedies}}&lt;/ref&gt; claiming that ''Simarouba'' is effective for treating certain diseases, there seems to be insufficient evidence&lt;ref&gt;{{Cite web|url=https://www.webmd.com/vitamins/ai/ingredientmono-371/simaruba|title=SIMARUBA: Overview, Uses, Side Effects, Precautions, Interactions, Dosing and Reviews}}&lt;/ref&gt; of curing diarrhea, malaria, edema, fever and stomach upset. Known in India as ''Lakshmi Taru'', the extracts from parts of the tree have been claimed to possess potent anticancer properties. However, to date,  no systematic  research using phytochemicals isolated from ''Simarouba glauca'' has been carried out to explore the molecular mechanisms leading to cancer cell death.&lt;ref name=Asha&gt;{{cite journal |last1=Asha Jose, Elango Kannan, Palur Ramakrishnan Anand Vijaya Kumar, SubbaRao Venkata Madhunapantula |title=Therapeutic Potential of Phytochemicals Isolated from Simarouba glauca for Inhibiting Cancers: A Review |journal=Systematic Reviews in Pharmacy|date=January–December 2019 |volume=10 |issue=1 |pages=73–80 |url=https://www.researchgate.net/publication/329712748 |accessdate=17 January 2020 |doi=10.5530/srp.2019.1.12}}&lt;/ref&gt; ''Simarouba'' extracts are known to be effective only on specific types of human cancer cell lines and tests conducted were invitro. Whether the same effect would be observed under invivo conditions, depends on bioavailability and bioaccessibility,&lt;ref name=Umesh&gt;{{cite journal |last1=T. G. Umesh |title=In-vitro antioxidant potential, free radical scavenging and cytotoxic activity of Simarouba gluaca leaves |journal=International Journal of Pharmacy and Pharmaceutical Sciences |date=2015 |volume=2 |pages=411–6 |s2cid=55638252 |url=https://web.archive.org/web/20200211225738/https://pdfs.semanticscholar.org/5b07/35bb85137521c26302efd869a4800ae562d1.pdf
|archive-date=2020-02-11 |accessdate=17 January 2020}}&lt;/ref&gt; hence ''Simarouba'' as an alternative cure for cancer remains unproven.
{{clear}}

==''Solanum virginianum''==
[[Image:Solanum Xanthocarpum.jpg|thumb|right|250px|Solanum Xanthocarpum is shown in Nepal. Credit: [[c:user:Krish Dulal|Krish Dulal]].{{tlx|free media}}]]
Family: ''Solanaceae''

The plant has many medical properties. In the tribes of Nilgiris, the plant is used to treat a whitlow (finger abscess): the finger is inserted into a ripe fruit for a few minutes.&lt;ref name=&quot;Tournebize&quot;&gt;Rémi Tournebize, [https://www.researchgate.net/publication/249656841_Points_on_the_ethno-ecological_knowledge_and_practices_among_four_Scheduled_Tribes_of_the_Nilgiris_Toda_Kota_Alu_Kurumba_and_Irula_with_emphasis_on_Toda_ethnobotany/file/72e7e51e67d305cd74.pdf ''Points on the ethno-ecological knowledge and practices among four Scheduled Tribes of the Nilgiris: Toda, Kota, Alu Kurumba and Irula, with emphasis on Toda ethnobotany''], Institute of Research for Development (Marseille), Thesis 2013, p. 103&lt;/ref&gt; In Nepal, a decoction of root is taken twice a day for seven days to treat cough, asthma and chest pain.&lt;ref name=Mahato&gt;RB Mahato, RP Chaudhary, [http://biologyeastborneo.com/wp-content/uploads/2011/07/Antibakteri-tumb.Nepal_.pdf ''Ethnomedicinal study and antibacterial activities of selected plants of Palpa district, Nepal''], Scientific World, Vol. 3, No. 3, July 2005, p. 29[4]&lt;/ref&gt;

Ayurvedic Physicians commonly used the drugs of Dashmula in their private practice. Dashmula comprises root of five trees (brihat  panchmula) and root of five small herbs (laghu  panchmula). Deep study in Ayurveda indicate that out of 33 species of ''Solanum'' from family ''Solanaceae'',  two species are used in Dashmula such  as ''Solanum anguivi'' Lam. (Bruhati) and ''Solanum virginianum'' L. (Kantkari) (Sharma, 2006). The tribals and villagers also used the drugs of Dashmula group for their common ailments. It is estimated that about 8000 metric  tons  of  roots of Dashmula are used annually by Ayurvedic industry in Maharashtra.&lt;ref name=Toro&gt;{{cite book |last1=Toro |first1=Dr. Sunita V. Toro |last2=Patil |first2=Dr. Anjali R. Patil |last3=Chavan |first3=Prof. (Dr.) N. S. Chavan |title=Floral wealth of Achara- A sacred village on central west coast of India |date=2013 |publisher=Dr. V. B. Helavi |pages=26–29 |url=https://www.researchgate.net/publication/321212997 |accessdate=13 February 2019 }}&lt;/ref&gt;{{rp|26}}

Heble ''et al.'', (1968) chemically isolated, crystallized, diosgenin and beta cytosterol constituents from Solanum virginianumL. Further they reported the presence of triterpenes like Tupeol. Heble ''et  al.'', (1971) noted the presence of coumarins, scopolin, scopoletin, esculin and esculetin from plant parts of ''Solanum virginianum'' through  column chromatography. Hussain ''et al.'', (2010) in addition to alkaloids content also determined the presence of flavoinoids and  saponin apart from the presence of tolerable level of heavy metals like Cu, Fe, Pb, Cd and Zn. Shankar ''et al.'', (2011) reported and quantified bioactive steroidal glycoalkaloid  khasianine  in  addition to solanine and solasomargine  through HPTLC. Apigenin showed antiallergic while diosgenin exhibited anti–inflammatory effects (Singh ''et al.'', 2010). The leaf extract inhibit the growth  of pathogenic organisms.(Seeba, 2009). Tanusak Changbanjong ''et al.'', (2010) reported the effect of crude extract of ''Solanum verginianum'' against snails and mosquito larvae.&lt;ref name=Toro/&gt;{{rp|28}}

''Solanum virginianum'' L. (Kantkari) is useful in bronchial asthma (Govindan ''et al.'', 1999). Krayer and Briggs (1950) reported the antiaccelerator cardiac action of solasodine and some of its  derivatives. The  plant possesses antiurolthiatic and natriuretic activities. (Patel ''et al.'', 2010). A decoction of the fruits of the  plant is used for treatment of diabetes (Nadkarni, 1954). ''Solanum virginianum'' L. herb is useful in cough, chest pain, against vomiting, hair fall, leprosy, itching scabies, skin diseases and cardiac diseases associated with edema (Kumar ''et al.'', 2010).&lt;ref name=Toro/&gt;{{rp|28}}

Roots decoction is used as fabrige, effective diuretic and expectorant. It is diuretic useful in the treatment of catarrhal, fever, cough, asthma, and chest pain (Ghani, 1996). Root paste is utilized  by the Mukundara tribals of Rajasthan for the treatment of hernia as well as in flatulence and constipation. Stem, flower and fruits are prescribed for relief in burning sensation in the feed. Leaves are applied locally to relieve body or muscle pains, while its juice mixed with black pepper is advised for rheumatism (Nadkarni, 1954). Fruit juice is useful in sore throats and rheumatism. A decoction of the fruits of the plant is used by tribal and rural people of Orissa for the treatment of diabetes (Nadkarni, 1954).&lt;ref name=Toro/&gt;{{rp|28}} Smoking the seeds of the dried solanum virginanum in a biri warp is said to allay toothache and tooth decay in Indian folk medicine.

''In-vitro'' antioxidant and ''in-vivo'' Antimutagenic properties of ''Solanum xanthocarpum'' seed extracts, the preliminary qualitative phytochemical screening was done which reveal the presence of polyphenols, flavonoids, glycoside, alkaloids, carbohydrates, and reducing sugar etc. Based preliminary qualitative phytochemical screening, Quantitative estimation of polyphenols was performed, quantitative estimation alcoholic extract found significant amounts of polyphenols as compare to aqueous extract. ''In-vitro'' antioxidants was performed by two method DDPH and superoxide radical scavenging method, the alcoholic extract shows significant antioxidant properties as compare to aqueous extract, based on polyphenols and antioxidant properties alcoholic extracts was used for the antimutagenic (clastogenic) test. Alcoholic extract produced significant result in antimutagenic activity.&lt;ref name=Vaidya&gt;Antioxidant and Antimutagenic (Anticlastogenic) Effect of ''Solanum xanthocarpum'' seed extracts. Santosh Kumar Vaidya, Dharmesh K. Golwala and Darpini S. Patel. International Journal of Pharmaceutical Sciences and Nanotechnology (ISSN: 0974-3278) 2020: Volume 13, Issue 4, page 5005-5010. [https://www.ijpsnonline.com/index.php/ijpsn/article/view/1048]&lt;/ref&gt;
{{clear}}

==''Stachys sieboldii''==
[[Image:Stachys sieboldii1.jpg|thumb|250px|right|Image shows ''Stachys sieboldii'' with red to purple flowers and reaching a height of 30 – 120 cm. Credit: Kurt Stueber.{{tlx|free media}}]]
Family: ''Lamiaceae''.

''Stachys affinis'', commonly called crosne, Chinese artichoke, Japanese artichoke, knotroot, or artichoke betony, is a perennial herbaceous plant of the family Lamiaceae, originating from China, with rhizomes that are a root vegetable that can be eaten raw, pickled, dried or cooked,&lt;ref name=Lim&gt;T.K. Lim, Edible Medicinal and Non-Medicinal Plants: Volume 11, Modifi ed Stems, Roots, Bulbs, DOI 10.1007/978-3-319-26062-4_3&lt;/ref&gt; where ''Stachys sieboldii'' is a synonym.

Vacuoles in the tuber of ''S. affinis'' are rich in stachyose.&lt;ref name=Greutert&gt;{{Cite journal|last1=Greutert|first1=H.|last2=Keller|first2=F.|date=1993-04-01|title=Further Evidence for Stachyose and Sucrose/H+ Antiporters on the Tonoplast of Japanese Artichoke (Stachys sieboldii) Tubers|journal=Plant Physiology
|volume=101|issue=4|pages=1317–1322|doi=10.1104/pp.101.4.1317|issn=0032-0889|pmid=12231787|pmc=160655}}&lt;/ref&gt; Stachyose is a tetrasaccharide, consist out of galactose, glucose and fructose. Stachyose is up to 80-90% in dry tubers.&lt;ref name=Yin&gt;{{Cite journal|last1=Yin|first1=J|last2=Yang|first2=G|last3=Wang|first3=S|last4=Chen|first4=Y|date=2006-08-15|title=Purification and determination of stachyose in Chinese artichoke (''Stachys Sieboldii'' Miq.) by high-performance liquid chromatography with evaporative light scattering detection|journal=Talanta|volume=70|issue=1|pages=208–212|doi=10.1016/j.talanta.2006.03.027|pmid=18970754|issn=0039-9140}}&lt;/ref&gt;

the entirety of ''S. affinis'' is used as an agent to treat colds and pneumonia.&lt;ref&gt;{{Cite journal|last1=Paton|first1=Alan|last2=Wu|first2=Zheng-yi|last3=Raven|first3=P. H.|date=1995|title=Flora of China Vol. 17: Verbenaceae through Solanaceae|journal=Kew Bulletin|volume=50|issue=4|pages=838|doi=10.2307/4110257|issn=0075-5974|jstor=4110257}}&lt;/ref&gt;

Root extract of ''S. affinis'' has shown antimicrobial activity.&lt;ref&gt;{{Cite journal|date=2002-12-01|title=Antimicrobial activity of the hexane extract of Stachys sieboldii MIQ leaf|journal=Journal of Life Science|volume=12|issue=6|pages=803–811|doi=10.5352/jls.2002.12.6.803|issn=1225-9918 }}&lt;/ref&gt; Antioxidant activity has been observed, plus inhibitory effects on acetylcholine esterase, monoamine oxidase and xanthine oxidase activities were observed in rat brains after 20 days feeding with methanolic extracts of ''S. affinis.''&lt;ref name=Antioxidant&gt;{{Cite journal|date=2004-02-01|title=Antioxidant Activities of Stachys sieboldii MIQ Roots|journal=Journal of Life Science|volume=14|issue=1|pages=1–7|doi=10.5352/jls.2004.14.1.001|issn=1225-9918 }}&lt;/ref&gt; Ethanol extract from this plant also seems to have antitumour activity.&lt;ref name=Ryu&gt;{{Cite journal|authors=Ryu BH, Bg P, Song SK |date=2002|title=Antitumor effects of the hexane extract of Stachys Sieboldii|journal=Biotechnol Bioeng|volume=17|issue=6|pages=520–524}}&lt;/ref&gt;

&quot;Japanese artichoke [...] contain germacrene D, caryophyllene, cadinene. [...] methanolic tuber extract of Japanese artichoke, which contains glycosides, including acteoside and stachysosides C, significantly inhibits induced mortality from potassium cyanide poisoning in mice [19]. This extract inhibits hyaluronidase activity, has anti-inflammatory action, and is effective in kidney disease [15].&quot;&lt;ref name=Slobodianiuk&gt;{{ cite book
|author=Slobodianiuk, Liudmyla; Budniak, Liliia 2; Marchyshyn, Svitlana ; Demydiak, Olha
|title= INVESTIGATION OF THE ANTI-INFLAMMATORY EFFECT OF THE DRY EXTRACT FROM THE HERB OF STACHYS SIEBOLDII M
|publisher=Pharmacologyonline
|location=Italy
|date=2021
|volume=
|editor=
|pages=590-597
|url=https://pharmacologyonline.silae.it/files/archives/2021/vol2/PhOL_2021_2_A067_Slobodianiuk.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=6 January 2022 }}&lt;/ref&gt;

The &quot;methanol extract from the leaves and root tubers and ethanol extract from the root tubers of Stachys sieboldii have a pronounced antibacterial effect on the culture of Salmonella typhimurium. In addition, methanol extract from the leaves of Stachys sieboldii showed a significant antibacterial effect on the culture of Bacillus cereus. It is believed that the antibacterial effect of Japanese artichoke is associated with the total content of polyphenols and flavonoids contained in the plant and which are extracted with methanol and ethanol [20].&quot;&lt;ref name=Slobodianiuk/&gt;
{{clear}}

==''Stenocereus queretaroensis''==

Family: ''Cactaceae''.

''Stenocereus queretaroensis'' is a species of cactus from Mexico, including the state of Querétaro, cultivated for its fruit.&lt;ref name=Ande01p647&gt;{{ cite book |last=Anderson |first=Edward F. |date=2001 |title=The Cactus Family |location=Pentland, Oregon |publisher=Timber Press |{{isbn|978-0-88192-498-5}} }}, pp. 647&lt;/ref&gt;

==''Taxandria parviceps''==

Family: ''Myrtaceae''.

''Taxandria parviceps'', commonly known as tea tree,&lt;ref name=angus&gt;&gt;{{cite web|url=http://www.gardeningwithangus.com.au/taxandria-parviceps-tea-tree/|title=''Taxandria parviceps'' – Tea Tree|accessdate=28 December 2016|publisher=Gardening With Angus}}&lt;/ref&gt; is a shrub species that grows on the south west coast of Western Australia.&lt;ref name=FloraBase&gt;&quot;Taxandria parviceps&quot;. FloraBase. Western Australian Government Department of Parks and Wildlife.&lt;/ref&gt; This plant was previously classified as ''Agonis parviceps'' but is now part of the ''Taxandria'' genus.

==''Theobroma cacao''==

Family: ''Malvaceae''.

Cocoa contains various phytochemicals, such as flavanols (including epicatechin), procyanidins, and other flavanoids. A systematic review presented moderate evidence that the use of flavanol-rich chocolate and cocoa products causes a small (2 mmHg) blood pressure lowering effect in healthy adults—mostly in the short term.&lt;ref&gt;{{Cite journal|last=Ried|first=Karin|last2=Fakler|first2=Peter|last3=Stocks|first3=Nigel P|date=2017-04-25|title=Effect of cocoa on blood pressure|url=https://doi.org/10.1002/14651858.CD008893.pub3|journal=Cochrane Database of Systematic Reviews|doi=10.1002/14651858.cd008893.pub3|issn=1465-1858|pmc=6478304|pmid=28439881}}&lt;/ref&gt;

The highest levels of cocoa flavanols are found in raw cocoa and to a lesser extent, dark chocolate, since flavonoids degrade during cooking used to make chocolate.&lt;ref&gt;{{cite news| url=http://news.bbc.co.uk/2/hi/health/6430777.stm | work=BBC News | title=Cocoa nutrient for 'lethal ills' | date=11 March 2007 | accessdate=30 April 2010}}&lt;/ref&gt; Cocoa also contains the stimulant compounds theobromine and caffeine. The beans contain between 0.1% and 0.7% caffeine, whereas dry coffee beans are about 1.2% caffeine.&lt;ref&gt;{{cite journal|pmid=24580540|date=2014|title=Cocoa phytochemicals: Recent advances in molecular mechanisms on health|journal=Critical Reviews in Food Science and Nutrition|volume=54|issue=11|pages=1458–72 |last1=Kim|first1=Jiyoung |last2=Kim|first2=Jaekyoon |last3=Shim|first3=J |last4=Lee|first4=CY |last5=Lee|first5=KW |last6=Lee|first6=HJ |doi=10.1080/10408398.2011.641041 |s2cid=20314911}}&lt;/ref&gt;

&quot;Cocoa is rich in polyphenols that have beneficial effects on cardiovascular disease.22 In cocoa, the polyphenols of particular interest are flavanols, a subclass of flavonoids, which are in turn a subclass of polyphenols. Cocoa is more than 10% flavanol by weight. Flavanols can be monomeric: in cocoa beans these are mainly (−)-epicatechin and (+)-catechin, dimeric (consisting of 2 units of epicatechin with differing linkages), or polymeric (combinations of monomers and chains of up to 10 units or more have been found). These polymers are known as procyanidins.1, 7, 16, 23, 24, 25, 26, 27, 28, 29, 30&quot;&lt;ref name=Drugs2021&gt;{{ cite web
|author=Drugs.com
|title= Cocoa
|publisher=Drugs.com
|location=
|date=5 January 2021
|url=https://www.drugs.com/npp/cocoa.html
|accessdate=16 September 2021 }}&lt;/ref&gt;

==''Tinospora cordifolia''==

Family ''Menispermaceae''.

Berberine is found in ''Tinospora cordifolia''.

==''Torreya californica''==

Family: ''Taxaceae''.

The California nutmeg, ''Torreya californica'', has a seed of similar appearance to nutmeg, but is not closely related to ''Myristica fragrans'', and is not used as a spice.

==''Tribulus terrestris''==

Family ''Zygophyllaceae''.

Phytochemicals of ''T. terrestris'' include steroidal saponins.&lt;ref&gt;{{ Cite journal|last=Dinchev|first=Dragomir|date=May 2007|title=Distribution of steroidal saponins in ''Tribulus terrestris'' from different geographical regions|journal=Phytochemistry|volume=69|issue=1|pages=176–186|pmid=17719068|doi=10.1016/j.phytochem.2007.07.003 }}&lt;/ref&gt;

==''Uncaria rhynchophylla''==
{{main|Remedy/Plants/Rubiaceae}}

==''Uncaria tomentosa''==
{{main|Remedy/Plants/Rubiaceae}}

==''Valeriana officinalis''==

Family ''Caprifoliaceae''.

===Alkaloids===

Actinidine,&lt;ref name=Shahidi&gt;{{ cite book
|author=Fereidoon Shahidi and Marian Naczk
|title=Phenolics in food and nutraceuticals
|location=Boca Raton, Florida, USA
|publisher=CRC Press
|url=https://web.archive.org/web/20130624105109/http://books.google.com/books?id=vHOJKw4umikC&amp;pg=PA313
|date=2013-06-24
|isbn=1-58716-138-9}}&lt;/ref&gt; chatinine,&lt;ref name=Shahidi/&gt;&lt;ref name=Waliszewski&gt;{{cite journal
|author=S. Waliszewski
|date=April 10, 1891
|title=Chatinine, alcaloïde de la racine de valériane
|journal=American Journal of Pharmacy
|volume=66
|url=https://web.archive.org/web/20130619055528/http://books.google.com/books?id=aPkKAAAAYAAJ&amp;pg=PA166 }}&lt;/ref&gt; shyanthine,&lt;ref name=Shahidi/&gt; valerianine,&lt;ref name=Shahidi/&gt; and valerine&lt;ref name=Shahidi/&gt;

===Flavanones===

Flavanones: hesperidin,&lt;ref name=&quot;pmid12895671&quot;&gt;{{cite journal |author = Marder M, Viola H, Wasowski C, Fernández S, Medina JH, Paladini AC |title = 6-methylapigenin and hesperidin: new valeriana flavonoids with activity on the CNS |journal = Pharmacol Biochem Behav |date=2003 |volume=75 |issue=3 |pages=537–45 |pmid=12895671 |doi=10.1016/S0091-3057(03)00121-7 }}&lt;/ref&gt; 6-methylapigenin,&lt;ref name=&quot;pmid12895671&quot; /&gt; and linarin&lt;ref name=&quot;pmid14751470&quot;&gt;{{cite journal |author = Fernández S, Wasowski C, Paladini AC, Marder M |title = Sedative and sleep-enhancing properties of linarin, a flavonoid-isolated from Valeriana officinalis |journal=Pharmacol Biochem Behav |date=2004 |volume=77 |issue=2 |pages=399–404 |pmid=14751470 |doi=10.1016/j.pbb.2003.12.003 }}&lt;/ref&gt; occur in Valerian.

==''Veratrum grandiflorum''==

''Veratrum'' is a genus of flowering plants in the family ''Melanthiaceae''.&lt;ref&gt;[http://www.tropicos.org/Name/40023758 Tropicos, ''Veratrum'' L.]&lt;/ref&gt; It occurs in damp habitats across much of temperate and subarctic Europe, Asia, and North America.&lt;ref name=e&gt;[http://apps.kew.org/wcsp/namedetail.do?name_id=291270 Kew World Checklist of Selected Plant Families]&lt;/ref&gt;&lt;ref&gt;[http://www.efloras.org/florataxon.aspx?flora_id=1&amp;taxon_id=134473 Flora of North America, Vol. 26 Page 72, False hellebore, skunk-cabbage, corn-lily, vérâtre, varaire, ''Veratrum'' Linnaeus, Sp. Pl. 2: 1044. 1753; Gen. Pl. ed. 5: 468. 1754. ]&lt;/ref&gt;&lt;ref&gt;[http://www.efloras.org/florataxon.aspx?flora_id=2&amp;taxon_id=134473 Flora of China Vol. 24 Page 82 ''Veratrum'' Linnaeus, Sp. Pl. 2: 1044. 1753.]&lt;/ref&gt;&lt;ref&gt;[http://luirig.altervista.org/flora/taxa/floraspecie.php?genere=Veratrum Altervista Flora Italiana, genere ''Veratrum''] includes photos and European distribution maps&lt;/ref&gt;&lt;ref&gt;[http://bonap.net/NAPA/TaxonMaps/Genus/County/Veratrum Biota of North America Program 2013 county distribution maps]&lt;/ref&gt;

''Veratrum'' species are vigorous herbaceous perennials with highly poisonous black rhizomes, and panicles of white or brown flowers on erect stems.&lt;ref name=RHSAZVeratrum&gt;{{cite book|title=RHS A–Z encyclopedia of garden plants|date=2008|publisher=Dorling Kindersley|location=United Kingdom|isbn=978-1405332965|pages=1136}}&lt;/ref&gt; In English they are known as both false hellebores and corn lilies.  However, ''Veratrum'' is not closely related to hellebores, corn, or lilies.

&quot;First isolated from ''Veratrum grandiflorum'' by Takaoka in the 1940s [19], [resveratrol (RSV, 3,4′,5-trihydroxystilbene)] RSV is found in food sources such as fruits, vegetables, and chocolate, and is better known as a constituent of grapes and wines, although it is present in only minimal quantities [18,20]. Due to its presence in wine, RSV attracted attention in the early 1990s to explain &quot;the French paradox&quot;, which suggested that people from France had a lower incidence of cardiovascular disease despite their high intake of saturated fats, presumably as a result of moderate red wine consumption [21].&quot;&lt;ref name=Coutinho&gt;{{ cite journal
|author=Diego de Sá Coutinho, Maria Talita Pacheco, Rudimar Luiz Frozza, and Andressa Bernardi
|title=Anti-Inflammatory Effects of Resveratrol: Mechanistic Insights
|journal=International Journal of Molecular Sciences
|date=20 June 2018
|volume=19
|issue=6
|pages=1812-1837
|url=https://www.mdpi.com/1422-0067/19/6/1812/pdf
|arxiv=
|bibcode=
|doi=10.3390/ijms19061812
|pmid=
|accessdate=16 September 2021 }}&lt;/ref&gt;

==''Viburnum lentago''==
{{main|Remedy/Plants/Adoxaceae}}

==''Vitis vinifera''==
[[Image:Cabernet Sauvignon Gaillac.jpg|thumb|right|250px|This bunch of grapes are of the Cabernet Sauvignon variety. Credit: [[:de:user:BerndtF|BerndtF]].{{tlx|free media}}]]
Family: ''Vitaceae''.

''V. vinifera'' contains many phenolic compounds.&lt;ref name=Olaizola&gt;{{Cite journal |last1=Aizpurua-Olaizola |first1=Oier |last2=Ormazabal |first2=Markel |last3=Vallejo |first3=Asier |last4=Olivares |first4=Maitane |last5=Navarro |first5=Patricia |last6=Etxebarria |first6=Nestor |last7=Usobiaga |first7=Aresatz |date=2015-01-01 |title=Optimization of Supercritical Fluid Consecutive Extractions of Fatty Acids and Polyphenols from Vitis Vinifera Grape Wastes |journal=Journal of Food Science |language=en |volume=80 |issue=1 |pages=E101–E107 |doi=10.1111/1750-3841.12715|pmid=25471637 |issn=1750-3841}}&lt;/ref&gt; Anthocyanins can be found in the skin of the berries, hydroxycinnamic acids in the pulp and condensed tannins of the proanthocyanidins type in the seeds. Stilbenoids can be found in the skin and in wood.

''Trans''-resveratrol is a phytoalexin produced against the growth of fungal pathogens such as ''Botrytis cinerea''&lt;ref name=Favaron&gt;{{cite journal |url=http://www.sipav.org/main/jpp/volumes/0309/030909.pdf |title=The role of grape polyphenols on trans-resveratrol activity against Botrytis cinerea and of fungal laccase on the solubility of putative grape PR proteins |first1=F. |last1=Favaron |first2=M. |last2=Lucchetta |first3=S. |last3=Odorizzi |first4=A. T. |last4=Pais da Cunha |first5=L. |last5=Sella |journal=Journal of Plant Pathology |year=2009 |volume=91 |issue=3 |pages=579–588 |doi=10.4454/jpp.v91i3.549 }}&lt;/ref&gt; and delta-viniferin is another grapevine phytoalexin produced following fungal infection by ''Plasmopara viticola''.&lt;ref name=Timperio&gt;{{cite journal
| last1=Timperio |first1=A. M.
| last2=d’Alessandro |first2=A.
| last3=Fagioni |first3=M.
| last4=Magro |first4=P.
| last5=Zolla |first5=L.
| title=Production of the phytoalexins trans-resveratrol and delta-viniferin in two economy-relevant grape cultivars upon infection with Botrytis cinerea in field conditions
| doi=10.1016/j.plaphy.2011.07.008
| journal=Plant Physiology and Biochemistry
| volume=50
| issue=1
| pages=65–71
| year=2012
| pmid=21821423 }}&lt;/ref&gt;
* Astringin

''Vitis vinifera'' red cultivars are rich in anthocyanins that impart their colour to the berries (generally in the skin). The 5 most basic anthocyanins found in grape are:
* Cyanidin-3-O-glucoside
* Delphinidin-3-O-glucoside
* Malvidin-3-O-glucoside
* Petunidin-3-O-glucoside
* Peonidin-3-O-glucoside

Cultivars like Graciano&lt;ref name=Nunez&gt;{{cite journal |last1=Núñez |first1=V. |last2=Monagas |first2=M. |last3=Gomez-Cordovés |first3=M. C. |last4=Bartolomé |first4=B. |title=Vitis vinifera L. Cv. Graciano grapes characterized by its anthocyanin profile |doi=10.1016/S0925-5214(03)00140-6 |journal=Postharvest Biology and Technology |volume=31 |pages=69–79 |year=2004}}&lt;/ref&gt;&lt;ref name=Monagas&gt;{{cite journal |title=Anthocyanin-derived Pigments in Graciano, Tempranillo, and Cabernet Sauvignon Wines Produced in Spain |first1=María |last1=Monagas |first2=Verónica |last2=Núñez |first3=Begoña |last3=Bartolomé |first4=Carmen |last4=Gómez-Cordovés |journal=Am. J. Enol. Vitic. |year=2003 |volume=54 |issue=3 |pages=163–169 |url=http://www.ajevonline.org/content/54/3/163.short}}&lt;/ref&gt; may also contain :
; acetylated anthocyanins
* Cyanidin-3-(6-acetyl)-glucoside
* Delphinidin-3-(6-acetyl)-glucoside
* Malvidin-3-(6-acetyl)-glucoside
* Petunidin-3-(6-acetyl)-glucoside
* Peonidin-3-(6-acetyl)-glucoside

; coumaroylated anthocyanins
* Cyanidin-3-(6-p-coumaroyl)-glucoside
* Delphinidin-3-(6-p-coumaroyl)-glucoside
* Malvidin-3-(6-p-coumaroyl)-glucoside cis
* Malvidin-3-(6-p-coumaroyl)-glucoside trans
* Petunidin-3-(6-p-coumaroyl)-glucoside
* Peonidin-3-(6-p-coumaroyl)-glucoside

; caffeoylated anthocyanins
* Malvidin-3-(6-p-caffeoyl)-glucoside
* Peonidin-3-(6-p-caffeoyl)-glucoside

Isoprenoid monoterpenes are present in grape, above all acyclic linalool, geraniol, nerol, citronellol, homotrienol and monocyclic α-terpineol, mostly occurring as glycosides. Carotenoids accumulate in ripening grape berries. Oxidation of carotenoids produces volatile fragments, C13-norisoprenoids. These are strongly odoriferous compounds, such as β-ionone (aroma of viola), damascenone (aroma of exotic fruits), β-damascone (aroma of rose) and β-ionol (aroma of flowers and fruits). Melatonin, an alkaloid, has been identified in grape.&lt;ref name=Iriti&gt;{{cite journal |pmid=19445314 |volume=4 |issue=5 |title=Bioactivity of grape chemicals for human health |journal=Natural Product Communications |date=May 2009 |pages=611–34 |last1=Iriti |first1=M |last2=Faoro |first2=F |doi=10.1177/1934578X0900400502 |s2cid=39638336 }}&lt;/ref&gt; In addition, seeds are rich in unsaturated fatty acids, which helps lowering levels of total cholesterol and LDL cholesterol in the blood.&lt;ref name=Olaizola/&gt;
{{clear}}

==''Withania somnifera''==

Family ''Solanaceae''.

The main phytochemical constituents in ashwagandha are withanolides &amp;ndash; which are triterpene lactones &amp;ndash; withaferin A, alkaloids, steroidal lactones, tropine, and cuscohygrine.&lt;ref name=&quot;drugsAsh&quot;&gt;{{cite web |title=Ashwagandha |url=https://www.drugs.com/npp/ashwagandha.html |publisher=Drugs.com |accessdate=2 February 2021 |date=2 November 2020}}&lt;/ref&gt; Some 40 withanolides, 12 alkaloids, and numerous sitoindosides have been isolated.&lt;ref name=drugsAsh/&gt; Withanolides are structurally similar to the ginsenosides of ''Panax ginseng'', leading to a common name for ''W. somnifera'', &quot;Indian ginseng&quot;.&lt;ref name=drugsAsh/&gt;

==''Wollemia nobilis''==
[[Image:Ancient Wollemi pines.jpeg|right|thumb|300px|A NPWS firefighter looks up at one of the ancient Wollemi pines discovered in 1994 he has been sent in to protect. Credit: NPWS firefighter.{{tlx|fairuse}}]]
''Wollemia'' is a genus of coniferous trees in the family ''Araucariaceae''.

&quot;Desperate efforts by firefighters on the ground and in the air have saved the only known natural grove of the world-famous Wollemi pines from destruction during the record-breaking bushfires in NSW.&quot;&lt;ref name=Hannam&gt;{{ cite web
|author=Peter Hannam
|title=Incredible, secret firefighting mission saves famous 'dinosaur trees'
|publisher=The Sydney Morning Herald
|location=Sydney, Australia
|date=January 15, 2020
|url=https://www.smh.com.au/environment/conservation/incredible-secret-firefighting-mission-saves-famous-dinosaur-trees-20200115-p53rom.html?utm_source=Nature+Briefing&amp;utm_campaign=8cb774fce7-briefing-dy-20200115&amp;utm_medium=email&amp;utm_term=0_c9dfd39373-8cb774fce7-43855389
|accessdate=15 January 2020 }}&lt;/ref&gt;

&quot;The rescue mission involved water-bombing aircraft and large air tankers dropping fire retardant. Helicopters also winched specialist firefighters into the remote gorge to set up an irrigation system to increase the moisture content of the ground fuels to slow the advance of any fire.&quot;&lt;ref name=Hannam/&gt;

&quot;Wollemi National Park is the only place in the world where these trees are found in the wild and, with less than 200 left, we knew we needed to do everything we could to save them.&quot;&lt;ref name=Kean&gt;{{ cite web
|author=Matt Kean
|title=Incredible, secret firefighting mission saves famous 'dinosaur trees'
|publisher=The Sydney Morning Herald
|location=Sydney, Australia
|date=January 15, 2020
|url=https://www.smh.com.au/environment/conservation/incredible-secret-firefighting-mission-saves-famous-dinosaur-trees-20200115-p53rom.html?utm_source=Nature+Briefing&amp;utm_campaign=8cb774fce7-briefing-dy-20200115&amp;utm_medium=email&amp;utm_term=0_c9dfd39373-8cb774fce7-43855389
|accessdate=15 January 2020 }}&lt;/ref&gt;

&quot;Fossil evidence indicates that the trees existed between 200 and 100 million years ago and were once present across the whole of Australia.&quot;&lt;ref name=Brack&gt;{{ cite web
|author=Cris Brack
|title=Incredible, secret firefighting mission saves famous 'dinosaur trees'
|publisher=The Sydney Morning Herald
|location=Sydney, Australia
|date=January 15, 2020
|url=https://www.smh.com.au/environment/conservation/incredible-secret-firefighting-mission-saves-famous-dinosaur-trees-20200115-p53rom.html?utm_source=Nature+Briefing&amp;utm_campaign=8cb774fce7-briefing-dy-20200115&amp;utm_medium=email&amp;utm_term=0_c9dfd39373-8cb774fce7-43855389
|accessdate=15 January 2020 }}&lt;/ref&gt;
{{clear}}

==''Xanthorhiza simplicissima''==

Family: ''Ranunculaceae''.

Berberine is found ''Xanthorhiza simplicissima'' (yellowroot).&lt;ref&gt;{{cite journal | author = Zhang Q, Cai L, Zhong G, Luo W | title = Simultaneous determination of jatrorrhizine, palmatine, berberine, and obacunone in Phellodendri Amurensis Cortex by RP-HPLC | journal = Zhongguo Zhong Yao Za Zhi = Zhongguo Zhongyao Zazhi = China Journal of Chinese Materia Medica | volume = 35 | issue = 16 | pages = 2061–4 | date = 2010 | pmid = 21046728 | doi = 10.4268/cjcmm20101603 }}&lt;/ref&gt;

==''Zanthoxylum''==

Family Rutaceae.

Historically, ''Zanthoxylum'' (Prickly ash) bark was used in traditional medicine.&lt;ref name =&quot;Wilbur&quot; &gt;Wilbur, C. Keith, MD. ''Revolutionary Medicine 1700-1800''. The Globe Pequot Press. Page 23. 1980.&lt;/ref&gt;

Species identified in Nigeria contains several types of alkaloids including benzophenanthridines (nitidine, dihydronitidine, oxynitidine, fagaronine, dihydroavicine, chelerythrine, dihydrochelerythrine, methoxychelerythrine, norchelerythrine, oxychelerythrine, decarine and fagaridine), furoquinolines (dictamine, 8-methoxydictamine, skimmianine, 3-dimethylallyl-4-methoxy-2-quinolone), carbazoles (3-methoxycarbazole, glycozoline), aporphines (berberine, tembetarine,&lt;ref name=Kanaya&gt;{{Cite web |title=Tembetarine |accessdate=2017-05-18 |url=https://web.archive.org/web/20171217135728/http://kanaya.naist.jp/knapsack_jsp/information.jsp?word=C00025347 }}&lt;/ref&gt; magnoflorine, M-methyl-corydine), canthinones (6-canthinone), acridones (1-hydroxy-3-methoxy-10-methylacridon-9-one, 1-hydroxy-10-methylacridon-9-one, zanthozolin), and aromatic and aliphatic amides.&lt;ref&gt;The Nigerian Zanthoxylum; Chemical and biological values. S. K. Adesina, Afr. J. Trad. CAM, 2005, volume 2, issue 3, pages 282-301 https://web.archive.org/web/20160303183649/https://tspace.library.utoronto.ca/bitstream/1807/9214/1/tc05032.pdf 2016-03-03&lt;/ref&gt; Hydroxy-alpha sanshool is a bioactive component of plants from the genus ''Zanthoxylum'', including the Sichuan pepper.

===Aporphines===

Aporphines include berberine and tembetarine found in Prickly ash bark.&lt;ref name=Kanaya/&gt;

===Benzophenanthridines===

Benzophenanthridines incude nitidine, dihydronitidine, oxynitidine, fagaronine, dihydroavicine, chelerythrine, dihydrochelerythrine, methoxychelerythrine, norchelerythrine, oxychelerythrine, decarine and fagaridine found in Prickly ash bark.&lt;ref name=Kanaya/&gt;

===Carbazoles===

Carbazoles include 3-methoxycarbazole and glycozoline found in Prickly ash bark.&lt;ref name=Kanaya/&gt;

===Furoquinolines===

Furoquinolines include dictamine, 8-methoxydictamine, skimmianine, and 3-dimethylallyl-4-methoxy-2-quinolone found in Prickly ash bark.&lt;ref name=Kanaya/&gt;

==''Zingiber mioga''==

Family:	''Zingiberaceae''.

==''Zingiber officinale''==

Family:	''Zingiberaceae''.

The characteristic fragrance and flavor of ginger result from volatile essential oil that compose 1-3% of the weight of fresh ginger, primarily consisting of sesquiterpenes, beta-bisabolene and zingiberene, zingerone, shogaols, and gingerols with [6]-gingerol (1-[4'-hydroxy-3'-methoxyphenyl]-5-hydroxy-3-decanone) as the major pungent compound.&lt;ref name=Ginger/&gt;&lt;ref name=An&gt;{{cite journal
|authors=An K, Zhao D, Wang Z, Wu J, Xu Y, Xiao G|year=2016|title=Comparison of different drying methods on Chinese ginger (Zingiber officinale Roscoe): Changes in volatiles, chemical profile, antioxidant properties, and microstructure|journal=Food Chemistry|volume=197|issue=Part B|pages=1292–300|doi=10.1016/j.foodchem.2015.11.033|pmid=26675871 }}&lt;/ref&gt; Some 400 chemical compounds exist in raw ginger.&lt;ref name=Ginger/&gt;

Zingerone is produced from gingerols during drying, having lower pungency and a spicy-sweet aroma.&lt;ref name=An/&gt; Shogaols are more pungent formed from gingerols during heating, storage or via acidity.&lt;ref name=Ginger&gt;{{cite web |title=Ginger |url=https://www.drugs.com/npp/ginger.html |publisher=Drugs.com |accessdate=25 November 2021 |date=20 December 2020}}&lt;/ref&gt;&lt;ref name=An/&gt; Numerous monoterpenes, amino acids, dietary fiber, protein, phytosterols, vitamins, and dietary minerals are other constituents.&lt;ref name=Ginger/&gt; Fresh ginger also contains an enzyme zingibain which is a cysteine protease and has similar properties to rennet.

Ginger could decrease body weight in obese subjects and increase HDL-cholesterol.&lt;ref name=Maharlouei&gt;{{cite journal|author=Maharlouei N, Tabrizi R, Lankarani KB, Rezaianzadeh A, Akbari M, Kolahdooz F, Rahimi M, Keneshlou F, Asemi Z.|date=2019|title=The effects of ginger intake on weight loss and metabolic profiles among overweight and obese subjects: A systematic review and meta-analysis of randomized controlled trials|journal=Critical Reviews in Food Science and Nutrition|url=https://www.tandfonline.com/doi/abs/10.1080/10408398.2018.1427044|volume=59|issue=11
|pages=1753–1766|doi=10.1080/10408398.2018.1427044|pmid=29393665|s2cid=35645698 }}&lt;/ref&gt;

==''Zingiber zerumbet''==

Family:	''Zingiberaceae''.

==See also==
{{div col|colwidth=20em}}
* [[Agronomy]]
* [[Remedy/Alcohols|Alcohols]]
* [[Remedy/Alkaloids|Alkaloids]]
* [[Allergies]]
* [[Aquaculture]]
* [[Remedy/Plants/Asteraceae|Asteraceae]]
* [[Remedy/Edema|Edema]]
* [[Remedy/Plants/Fabaceae|Fabaceae]]
* [[Remedy/Flavonoids|Flavonoids]]
* [[Forestry]]
* [[Fruit and its importance]]
* [[Gene project]]
* [[Reproductive health/Glandular system|Glandular system]]
* [[Genes/Expressions/Hair colors|Hair colors]]
* [[Horticulture]]
* [[Human skin pigmentation]]
* [[w:List of plant family names with etymologies|List of plant family names]]
* [[Remedy/Medicine|Medicinal remedies]]
* [[Medicine]]
* [[Remedy/Minerals|Minerals]]
* [[Remedy/Nutraceuticals|Neutraceuticals]]
* [[Remedy/Oils|Oils]]
* [[Oral Medicine and Oral Pathology/Pigmented lesions of the oromucosa|Pigmented lesions of the oromucosa]]
* [[Remedy/Plants|Plants]]
* [[Remedy/Polyphenols|Polyphenols]]
* [[Pomology]]
* [[Remedy/Plants/Rubiaceae|Rubiaceae]]
* [[Remedy/Terpenoids|Terpenoids]]
* [[Remedy/Vitamins|Vitamins]]
* [[Remedy/Waxes|Waxes]]
{{Div col end}}

==References==
{{reflist|2}}

==External links==

&lt;!-- footer templates --&gt;
{{Medicine resources}}{{Sisterlinks|Medicinal plants}}

&lt;!-- categories --&gt;
[[Category:Agronomy]]
[[Category:Biochemistry]]
[[Category:Horticulture]]
[[Category:Humanities]]
[[Category:Immunology]]
[[Category:Psychiatry]]
[[Category:Vitamins]]</text>
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  <page>
    <title>Maritime Health Research and Education-NET/The International Type 2 Diabetes Mellitus and Hypertension Research Group</title>
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      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
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      <text bytes="4381" sha1="dci1ww0h95kgon15u573hygdwlb2seo" xml:space="preserve">==International Prediabetes,Diabetes and Hypertension Research Group==

The International Diabetes and Hypertension Research Group for fishers, seafarers and other transport workers was created the 12 Jan 2022 in a Zoom conference by specialists in diabetes epidemiology, diabetology, occupational epidemiology, occupational/maritime medicine and public health from Denmark, Greenland, Spain, France, Panamá. Russia and The Filippines. &lt;ref&gt;https://portal.findresearcher.sdu.dk/en/projects/the-international-diabetes-and-hypertension-research-group-in-sea&lt;/ref&gt;The aim is to provide a foundation for safe and healthy preventive strategies within the UN Global Sustainable Goals, especially '''Goal 3:''' Good health and well-being for all workers ,'''Goal 4:''' Quality education,'''Goal 8:''' Decent Work and Economic Growth and '''Goal 17''': Partnerships to achieve the goals with the primary tasks. Millions of medical examinations are done every year for seafarers, fishers, truck drivers, loco-, bus- and taxi drivers. Most of them use the non-valid urine-sticks and no valid test for T2D, no A1c or FG. By adding biannual screening of hypertension and diabetes mellitus the target groups can be rescued from loss of workability, loss of QUALies and loss of living years. Screening for hypertension and diabetes is cost-effective and sustainable with low extra cost, possibility for no or small extra visit fee and A1C test around 20-50 USD&lt;ref&gt;https://www.talktomira.com/post/what-is-a-diabetes-screening-test-and-how-much-it-costs&lt;/ref&gt; as the target group need to attend to the medical clinics for their obligatory often biannual medical examination, anyway. While the extra cost for A1c blood test, lack of laboratory access and clinical time consume might be a problem, then the strategy is to validate and replace A1C with Glukometer test&lt;ref&gt;Chen, Huizhen, Qingtao Yao, Yang Dong, Zhimei Tang, Ruiying Li, Baochao Cai, Ruili Wang, and Qiu Chen. “The Accuracy Evaluation of Four Blood Glucose Monitoring Systems According to ISO 15197:2003 and ISO 15197:2013 Criteria.” Primary Care Diabetes 13, no. 3 (June 2019): 252–58. https://doi.org/10.1016/j.pcd.2018.12.010&lt;/ref&gt;
&lt;ref&gt;Chubb, S. A. Paul, Kylie Van Minnen, Wendy A. Davis, David G. Bruce, and Timothy M. E. Davis. “The Relationship between Self-Monitoring of Blood Glucose Results and Glycated Haemoglobin in Type 2 Diabetes: The Fremantle Diabetes Study.” Diabetes Research and Clinical Practice 94, no. 3 (December 2011): 371–76. https://doi.org/10.1016/j.diabres.2011.07.038&lt;/ref&gt;
&lt;ref&gt;Kenning, Matthes, Anselm Puchert, Sabine Berg, and Eckhard Salzsieder. “System Accuracy of the Blood Glucose Monitoring System TD4216.” Journal of Diabetes Science and Technology 14, no. 5 (March 7, 2020): 976–77. https://doi.org/10.1177/1932296820910785.&lt;/ref&gt;
&lt;ref&gt;Makris, K., L. Spanou, A. Rambaouni-Antoneli, K. Koniari, I. Drakopoulos, D. Rizos, and A. Haliassos. “Relationship between Mean Blood Glucose and Glycated Haemoglobin in Type 2 Diabetic Patients.” Diabetic Medicine: A Journal of the British Diabetic Association 25, no. 2 (February 2008): 174–78. https://doi.org/10.1111/j.1464-5491.2007.02379.x.&lt;/ref&gt;
&lt;ref&gt;Pashintseva, L. P., V. S. Bardina, I. R. Il’iasov, B. P. Mishchenko, and M. B. Antsiferov. “[The clinical laboratory evaluation of accuracy of portable glucometers ‘Satellite Express’ and ‘Satellite Express mini’].” Klinicheskaia Laboratornaia Diagnostika, no. 11 (November 2011): 33–35.&lt;/ref&gt;
&lt;ref&gt;Sarwat, S., L. L. Ilag, M. A. Carey, D. S. Shrom, and R. J. Heine. “The Relationship between Self-Monitored Blood Glucose Values and Glycated Haemoglobin in Insulin-Treated Patients with Type 2 Diabetes.” Diabetic Medicine: A Journal of the British Diabetic Association 27, no. 5 (May 2010): 589–92. https://doi.org/10.1111/j.1464-5491.2010.02955.x.&lt;/ref&gt; The specific educational needs for the prevention of the diabetes type 2 and hypertension in the maritime area has never been described and a list of Workshops is planned for the next year as part of the T2D/HTN Research and Education plan 2023-2032

ccccc
==[[/Revision of the WHO International Medical Guide for Ships/]]==
==[[/Revision of the Ships Medical Chest/]]==
==[[/Members/]]==
== [[/Learning from Interventions in non-maritime workplaces/]]==

==References==
[[Category:Hypertension]]</text>
      <sha1>dci1ww0h95kgon15u573hygdwlb2seo</sha1>
    </revision>
    <revision>
      <id>2805747</id>
      <parentid>2805746</parentid>
      <timestamp>2026-04-21T12:36:34Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4374" sha1="ffpq1979o416l8dujgf0q0ynrvetwi3" xml:space="preserve">==International Prediabetes,Diabetes and Hypertension Research Group==

The International Diabetes and Hypertension Research Group for fishers, seafarers and other transport workers was created the 12 Jan 2022 in a Zoom conference by specialists in diabetes epidemiology, diabetology, occupational epidemiology, occupational/maritime medicine and public health from Denmark, Greenland, Spain, France, Panamá. Russia and The Filippines. &lt;ref&gt;https://portal.findresearcher.sdu.dk/en/projects/the-international-diabetes-and-hypertension-research-group-in-sea&lt;/ref&gt;The aim is to provide a foundation for safe and healthy preventive strategies within the UN Global Sustainable Goals, especially '''Goal 3:''' Good health and well-being for all workers ,'''Goal 4:''' Quality education,'''Goal 8:''' Decent Work and Economic Growth and '''Goal 17''': Partnerships to achieve the goals with the primary tasks. Millions of medical examinations are done every year for seafarers, fishers, truck drivers, loco-, bus- and taxi drivers. Most of them use the non-valid urine-sticks and no valid test for T2D, no A1c or FG. By adding biannual screening of hypertension and diabetes mellitus the target groups can be rescued from loss of workability, loss of QUALies and loss of living years. Screening for hypertension and diabetes is cost-effective and sustainable with low extra cost, possibility for no or small extra visit fee and A1C test around 20-50 USD&lt;ref&gt;https://www.talktomira.com/post/what-is-a-diabetes-screening-test-and-how-much-it-costs&lt;/ref&gt; as the target group need to attend to the medical clinics for their obligatory often biannual medical examination, anyway. While the extra cost for A1c blood test, lack of laboratory access and clinical time consume might be a problem, then the strategy is to validate and replace A1C with Glukometer test&lt;ref&gt;Chen, Huizhen, Qingtao Yao, Yang Dong, Zhimei Tang, Ruiying Li, Baochao Cai, Ruili Wang, and Qiu Chen. “The Accuracy Evaluation of Four Blood Glucose Monitoring Systems According to ISO 15197:2003 and ISO 15197:2013 Criteria.” Primary Care Diabetes 13, no. 3 (June 2019): 252–58. https://doi.org/10.1016/j.pcd.2018.12.010&lt;/ref&gt;
&lt;ref&gt;Chubb, S. A. Paul, Kylie Van Minnen, Wendy A. Davis, David G. Bruce, and Timothy M. E. Davis. “The Relationship between Self-Monitoring of Blood Glucose Results and Glycated Haemoglobin in Type 2 Diabetes: The Fremantle Diabetes Study.” Diabetes Research and Clinical Practice 94, no. 3 (December 2011): 371–76. https://doi.org/10.1016/j.diabres.2011.07.038&lt;/ref&gt;
&lt;ref&gt;Kenning, Matthes, Anselm Puchert, Sabine Berg, and Eckhard Salzsieder. “System Accuracy of the Blood Glucose Monitoring System TD4216.” Journal of Diabetes Science and Technology 14, no. 5 (March 7, 2020): 976–77. https://doi.org/10.1177/1932296820910785.&lt;/ref&gt;
&lt;ref&gt;Makris, K., L. Spanou, A. Rambaouni-Antoneli, K. Koniari, I. Drakopoulos, D. Rizos, and A. Haliassos. “Relationship between Mean Blood Glucose and Glycated Haemoglobin in Type 2 Diabetic Patients.” Diabetic Medicine: A Journal of the British Diabetic Association 25, no. 2 (February 2008): 174–78. https://doi.org/10.1111/j.1464-5491.2007.02379.x.&lt;/ref&gt;
&lt;ref&gt;Pashintseva, L. P., V. S. Bardina, I. R. Il’iasov, B. P. Mishchenko, and M. B. Antsiferov. “[The clinical laboratory evaluation of accuracy of portable glucometers ‘Satellite Express’ and ‘Satellite Express mini’].” Klinicheskaia Laboratornaia Diagnostika, no. 11 (November 2011): 33–35.&lt;/ref&gt;
&lt;ref&gt;Sarwat, S., L. L. Ilag, M. A. Carey, D. S. Shrom, and R. J. Heine. “The Relationship between Self-Monitored Blood Glucose Values and Glycated Haemoglobin in Insulin-Treated Patients with Type 2 Diabetes.” Diabetic Medicine: A Journal of the British Diabetic Association 27, no. 5 (May 2010): 589–92. https://doi.org/10.1111/j.1464-5491.2010.02955.x.&lt;/ref&gt; The specific educational needs for the prevention of the diabetes type 2 and hypertension in the maritime area has never been described and a list of Workshops is planned for the next year as part of the T2D/HTN Research and Education plan 2023-2030
==[[/Revision of the WHO International Medical Guide for Ships/]]==
==[[/Revision of the Ships Medical Chest/]]==
==[[/Members/]]==
== [[/Learning from Interventions in non-maritime workplaces/]]==

==References==
[[Category:Hypertension]]</text>
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        <id>1417466</id>
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      <format>text/x-wiki</format>
      <text bytes="4386" sha1="lfgtnio22vspvfbxha88vljj6v1ozt6" xml:space="preserve">==International Prediabetes,Diabetes and Hypertension Research Group==

The International Diabetes and Hypertension Research Group for fishers, seafarers and other transport workers was created the 12 Jan 2022 in a Zoom conference by specialists in diabetes epidemiology, diabetology, occupational epidemiology, occupational/maritime medicine and public health from Denmark, Greenland, Spain, France, Panamá. Russia and The Filippines. &lt;ref&gt;https://portal.findresearcher.sdu.dk/en/projects/the-international-diabetes-and-hypertension-research-group-in-sea&lt;/ref&gt;The aim is to provide a foundation for safe and healthy preventive strategies within the UN Global Sustainable Goals, especially '''Goal 3:''' Good health and well-being for all workers ,'''Goal 4:''' Quality education,'''Goal 8:''' Decent Work and Economic Growth and '''Goal 17''': Partnerships to achieve the goals with the primary tasks. Millions of medical examinations are done every year for seafarers, fishers, truck drivers, loco-, bus- and taxi drivers. Most of them use the non-valid urine-sticks and no valid test for T2D, no A1c or FG. By adding biannual screening of hypertension and diabetes mellitus the target groups can be rescued from loss of workability, loss of QUALies and loss of living years. Screening for hypertension and diabetes is cost-effective and sustainable with low extra cost, possibility for no or small extra visit fee and A1C test around 20-50 USD&lt;ref&gt;https://www.talktomira.com/post/what-is-a-diabetes-screening-test-and-how-much-it-costs&lt;/ref&gt; as the target group need to attend to the medical clinics for their obligatory often biannual medical examination, anyway. While the extra cost for A1c blood test, lack of laboratory access and clinical time consume might be a problem, then the strategy is to validate and replace A1C with Glukometer test&lt;ref&gt;Chen, Huizhen, Qingtao Yao, Yang Dong, Zhimei Tang, Ruiying Li, Baochao Cai, Ruili Wang, and Qiu Chen. “The Accuracy Evaluation of Four Blood Glucose Monitoring Systems According to ISO 15197:2003 and ISO 15197:2013 Criteria.” Primary Care Diabetes 13, no. 3 (June 2019): 252–58. https://doi.org/10.1016/j.pcd.2018.12.010&lt;/ref&gt;
&lt;ref&gt;Chubb, S. A. Paul, Kylie Van Minnen, Wendy A. Davis, David G. Bruce, and Timothy M. E. Davis. “The Relationship between Self-Monitoring of Blood Glucose Results and Glycated Haemoglobin in Type 2 Diabetes: The Fremantle Diabetes Study.” Diabetes Research and Clinical Practice 94, no. 3 (December 2011): 371–76. https://doi.org/10.1016/j.diabres.2011.07.038&lt;/ref&gt;
&lt;ref&gt;Kenning, Matthes, Anselm Puchert, Sabine Berg, and Eckhard Salzsieder. “System Accuracy of the Blood Glucose Monitoring System TD4216.” Journal of Diabetes Science and Technology 14, no. 5 (March 7, 2020): 976–77. https://doi.org/10.1177/1932296820910785.&lt;/ref&gt;
&lt;ref&gt;Makris, K., L. Spanou, A. Rambaouni-Antoneli, K. Koniari, I. Drakopoulos, D. Rizos, and A. Haliassos. “Relationship between Mean Blood Glucose and Glycated Haemoglobin in Type 2 Diabetic Patients.” Diabetic Medicine: A Journal of the British Diabetic Association 25, no. 2 (February 2008): 174–78. https://doi.org/10.1111/j.1464-5491.2007.02379.x.&lt;/ref&gt;
&lt;ref&gt;Pashintseva, L. P., V. S. Bardina, I. R. Il’iasov, B. P. Mishchenko, and M. B. Antsiferov. “[The clinical laboratory evaluation of accuracy of portable glucometers ‘Satellite Express’ and ‘Satellite Express mini’].” Klinicheskaia Laboratornaia Diagnostika, no. 11 (November 2011): 33–35.&lt;/ref&gt;
&lt;ref&gt;Sarwat, S., L. L. Ilag, M. A. Carey, D. S. Shrom, and R. J. Heine. “The Relationship between Self-Monitored Blood Glucose Values and Glycated Haemoglobin in Insulin-Treated Patients with Type 2 Diabetes.” Diabetic Medicine: A Journal of the British Diabetic Association 27, no. 5 (May 2010): 589–92. https://doi.org/10.1111/j.1464-5491.2010.02955.x.&lt;/ref&gt; The specific educational needs for the prevention of the diabetes type 2 and hypertension in the maritime area has never been described and a list of Workshops is planned for the next year as part of the T2D/HTN Research and Education plan 2023-2030


==[[/Revision of the WHO International Medical Guide for Ships/]]==
==[[/Revision of the Ships Medical Chest/]]==
==[[/Members/]]==
== [[/Learning from Interventions in non-maritime workplaces/]]==

==References==
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==References==
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      <text bytes="2584" sha1="3vxx5arv0n0khoism7dz1orax8w6t9d" xml:space="preserve">==The John Snow Institute==

www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026
www.Prediabetes-Remission.com

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026
2. Prevalence studies
The-International-Maritime-Health-Database
Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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        <username>Saltrabook</username>
        <id>1417466</id>
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      <text bytes="1828" sha1="tl7sn2zzv3cezk3si0by53mxvilevtr" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026
2. Prevalence studies
The-International-Maritime-Health-Database
Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
      <sha1>tl7sn2zzv3cezk3si0by53mxvilevtr</sha1>
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        <username>Saltrabook</username>
        <id>1417466</id>
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      <format>text/x-wiki</format>
      <text bytes="2017" sha1="d5atcu5klrc6nnn1uk60lowszywx1mj" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;


2. Prevalence studies
The-International-Maritime-Health-Database
Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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    <revision>
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      <format>text/x-wiki</format>
      <text bytes="1984" sha1="2g41lelwzluheg1p2trn78t0kosy9gn" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e &lt;/ref&gt;


2. Prevalence studies
The-International-Maritime-Health-Database
Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
      <sha1>2g41lelwzluheg1p2trn78t0kosy9gn</sha1>
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    <revision>
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      <format>text/x-wiki</format>
      <text bytes="2166" sha1="01yej282sd8cgo2wwq48ogeo8obnv8j" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://www.dropbox.com/scl/fi/ss8yciy4jamyj55oetlgi/WORD-Links-to-WIX-References.docx?cloud_editor=word&amp;dl=0&amp;rlkey=qdpsi0ygllro23stf5lq4wg6e &lt;/ref&gt;

&lt;ref&gt; &lt;/ref&gt;

2. Prevalence studies. 
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <format>text/x-wiki</format>
      <text bytes="2185" sha1="h6nniyjqz51evwtu56ih7yhrar2iwza" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;


2. Prevalence studies

The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

Nursing Students Health Database
Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <text bytes="2346" sha1="ggi031c94fcdn76js6b56j6omqwg9at" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

Medical student's Health Database
School childrens Health database
3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <format>text/x-wiki</format>
      <text bytes="2351" sha1="3xanrcrix522q81p0hxjhb0axharv0y" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

3. Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

Medical student's Health Database

School childrens Health database

3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <text bytes="2515" sha1="h2fn0f1wn64fgv6hz7a6o7ejzq6ytce" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

3. Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;


School childrens Health database

3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <text bytes="2518" sha1="0jdq5wkdeiic45naluwibgs2py9p09i" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

3. Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;


School childrens Health database

3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Hypertension]]</text>
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      <text bytes="2518" sha1="1nx2bwd95lfllil82za52qldps5xygv" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt;

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

3. Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;


School childrens Health database

3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
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      <text bytes="2517" sha1="7emqluxukbhyiccppxl4636e0sakto6" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. Prevalence studies
The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

3. Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

4. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

School childrens Health database

3. Intervention studies 
English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
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      <text bytes="2695" sha1="stb7app3ct8ackqr6qooef7qagrs2c1" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. Prevalence studies

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. Intervention studies 

English invitation;  Spanish invitation; Invitation to Danish
4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
      <sha1>stb7app3ct8ackqr6qooef7qagrs2c1</sha1>
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    <revision>
      <id>2805782</id>
      <parentid>2805773</parentid>
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      <text bytes="3101" sha1="9f1mai8s3y7m43b2ri8mc3ju634b8ly" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. Prevalence studies

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. Intervention studies 
english
https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx

Spanish 
https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv 

dansk
https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv 


4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
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      <id>2805783</id>
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      <text bytes="4250" sha1="cm32hdp0eprmra7mfjgo9f1ctl2ok5d" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. Prevalence studies

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. Intervention studies 


English https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt;

Spanish &lt;ref&gt;https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv&lt;/ref&gt; 

Danish &lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt;


3. Intervention studies 

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt;

Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt;

Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools 

&lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt;

John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

7. Prediabetes-Remission Research Network:


4. General research protocol draft
5. Health Promoting Schools
6. John Snow Institute

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
      <sha1>cm32hdp0eprmra7mfjgo9f1ctl2ok5d</sha1>
    </revision>
    <revision>
      <id>2805786</id>
      <parentid>2805783</parentid>
      <timestamp>2026-04-21T15:04:36Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <origin>2805786</origin>
      <model>wikitext</model>
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      <text bytes="3620" sha1="eew55vh6r3ckruw18r9n5opm7ceadjx" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. Prevalence studies

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. Intervention studies 

3. Intervention studies

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt;

Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt;

Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

7. Prediabetes-Remission Research Network:

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
      <sha1>eew55vh6r3ckruw18r9n5opm7ceadjx</sha1>
    </revision>
    <revision>
      <id>2805790</id>
      <parentid>2805786</parentid>
      <timestamp>2026-04-21T15:14:43Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <origin>2805790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3636" sha1="5sdb4edm6m4tcrniy67ixxo5rewft6s" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt;

Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt;

Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. '''General research protocol draft'''
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. '''Health Promoting Schools''' &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  '''John Snow Institute''' &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
      <sha1>5sdb4edm6m4tcrniy67ixxo5rewft6s</sha1>
    </revision>
    <revision>
      <id>2805793</id>
      <parentid>2805790</parentid>
      <timestamp>2026-04-21T16:22:55Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <origin>2805793</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3616" sha1="hp8asatci8qhm42y6in395nvsoql2fs" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]</text>
      <sha1>hp8asatci8qhm42y6in395nvsoql2fs</sha1>
    </revision>
    <revision>
      <id>2805796</id>
      <parentid>2805793</parentid>
      <timestamp>2026-04-21T16:34:33Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
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      <origin>2805796</origin>
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      <format>text/x-wiki</format>
      <text bytes="3967" sha1="d820aa6kzrh87v2hcp8rmp2fpe3wy97" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives</text>
      <sha1>d820aa6kzrh87v2hcp8rmp2fpe3wy97</sha1>
    </revision>
    <revision>
      <id>2805870</id>
      <parentid>2805796</parentid>
      <timestamp>2026-04-22T05:26:02Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <minor />
      <origin>2805870</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3968" sha1="rsdu7frvz96clxs5l2s2ahx9qntniry" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY.

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives</text>
      <sha1>rsdu7frvz96clxs5l2s2ahx9qntniry</sha1>
    </revision>
    <revision>
      <id>2805871</id>
      <parentid>2805870</parentid>
      <timestamp>2026-04-22T05:41:24Z</timestamp>
      <contributor>
        <username>Saltrabook</username>
        <id>1417466</id>
      </contributor>
      <origin>2805871</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3967" sha1="d820aa6kzrh87v2hcp8rmp2fpe3wy97" xml:space="preserve">==The John Snow Institute==

INVITATION TO THE PREDIABETES-REMISSION STUDY

Billions are at increased risk of developing metabolic syndromes like prediabetes, type 2 diabetes and hypertension with significant social bias. It’s not just that people need more education; it’s that the environments they live in often make healthy choices harder than unhealthy ones.
The 16-weeks educational course include self-monitoring of blood sugar with glucometer, blood pressure, weight, and self-evaluation of diet and physical activity. The target is globally, but especially in the low- and middle-income countries where the availability of essential diagnostics remains a challenge.

Links:

1. Publications and pptx 2016-2026 &lt;ref&gt;https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute &lt;/ref&gt; 

2. '''Prevalence studies'''

2.1 The-International-Maritime-Health-Database &lt;ref&gt;https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=pt0kdesvmagcxaa2wez3tmza3 &lt;/ref&gt;

2.2 Nursing Students Health Database &lt;ref&gt; https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=onbjh4o8ko1lzdvgyi8nlrotk &lt;/ref&gt;

2.3. Medical student's Health Database &lt;ref&gt;https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=xyfqen5trdc5lniaovipl548n &lt;/ref&gt;

2.4. School childrens Health database &lt;ref&gt; https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&amp;dl=0&amp;rlkey=zlyz5wn673wf7owettq3nx3h5 &lt;/ref&gt;

3. '''Intervention studies'''

Englsh 
&lt;ref&gt;https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&amp;dl=0&amp;rlkey=7kzg91tqfgjskxf5aji8khicx &lt;/ref&gt; Danish
&lt;ref&gt;https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=x63w8oqvarz284zg2btq2johv &lt;/ref&gt; Spanish &lt;ref&gt; https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&amp;dl=0&amp;rlkey=popmr1fnodh1v951v9l7k9ezv &lt;/ref&gt;

4. General research protocol draft
&lt;ref&gt; https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&amp;dl=0&amp;rlkey=wat63e25ritmujwcpss8s4v0s &lt;/ref&gt;

5. Health Promoting Schools &lt;ref&gt; https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&amp;dl=0&amp;rlkey=673jyzcmwbfw7k9ui9nmtp0zh &lt;/ref&gt; 

6  John Snow Institute &lt;ref&gt; https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&amp;dl=0&amp;rlkey=lz2gi7mslcoay5dzygg8h6n6r &lt;/ref&gt;

'''7.''' '''Prediabetes-Remission Research Network:'''

Prof. Ing. MSc. Nailet Delgado; Prof Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,

==References==
[[Category:Prediabetes ]]

Education 1: Research Methodology
Education 3: The Health Journal Club
Education 4: Effectiveness of training in prevention for type 2 diabetes
Questionnaire Based studies: Protocols and Questionnaires
Systematic Review Studies
Organisation
Presentations
Invitations for collaboration
Consortium for Maritime Health Research and Education
Objectives</text>
      <sha1>d820aa6kzrh87v2hcp8rmp2fpe3wy97</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2805771</id>
      <parentid>2805641</parentid>
      <timestamp>2026-04-21T13:49:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2805771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="e3bp0k3ibu3hxb2tqs722igevbthiqb" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260421.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>e3bp0k3ibu3hxb2tqs722igevbthiqb</sha1>
    </revision>
    <revision>
      <id>2805905</id>
      <parentid>2805771</parentid>
      <timestamp>2026-04-22T07:11:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2805905</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="505zg9d5qy0dura5nayxh9j8pkx5v6c" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260422.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>505zg9d5qy0dura5nayxh9j8pkx5v6c</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods/Encyrtidae</title>
    <ns>0</ns>
    <id>299742</id>
    <revision>
      <id>2805780</id>
      <parentid>2805536</parentid>
      <timestamp>2026-04-21T14:23:09Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Encyrtidae - genera */</comment>
      <origin>2805780</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4683" sha1="pbggi8jeyjvfbbmr2sh4eyzci7wm0o1" xml:space="preserve">There are more than 640 described species of Afrotropical Encyrtidae in about 130 genera. More than 380 of these species have been found in South Africa.

==Diagnostic features of Encyrtidae==
In macro photographs, encyrtids are often recognizable using these features:&lt;br&gt;
# Large mesopleuron, usually covering more than half of the thorax (mesosoma) in side view.
# Mid coxae join the thorax near the middle of the mesopleuron.
# Mesoscutum transverse (width greater than length), and generally not segmented by notauli.
# The axillae are usually visible as triangles with two long sides, one adjacent to the mesoscutum and the other adjacent to the scutellum; the short side of the triangle is adjacent to the base of the forewing. The axillae touch, or nearly touch, medially; appearing as wedges between the mesoscutum and the scutellum.
# Cercal plates (at the base of the cercal bristles) are advanced; rather than being near the posterior tip of the metasoma (abdomen); they are usually within the anterior (front) two-thirds of the metasoma.
#In Encyrtidae with fully developed wings, the marginal vein of the forewing is usually shorter than stigmal vein; there is also an oblique band on the forewing that lacks setae (a linea calva).
&lt;gallery mode=packed heights=300&gt;
Encyrtidae lateral view with annotations.jpg
Encyrtidae dorsal view with annotations.jpg
Encyrtidae dorso-lateral view with annotations.jpg
Forewing of Anagyrus minipedicellus from Malaysia Fig7 Zu et al 2018.jpg
&lt;/gallery&gt;

== Encyrtidae - genera ==
An illustrated key to the encyrtid genera of the Afrotropics was published in 1979.&lt;ref name=Prinsloo&amp;Anneke1979&gt;Prinsloo, G. L., &amp; Annecke, D. P. (1979). A key to the genera of Encyrtidae from the Ethiopian region, with descriptions of three new genera (Hymenoptera: Chalcidoidea). Journal of the Entomological Society of southern Africa, 42(2), 349-382. [https://journals.co.za/doi/pdf/10.10520/AJA00128789_2641 PDF]&lt;/ref&gt;. This key has been updated [[African Arthropods/Afrotropical Encyrtidae Key|'''here''']] to include some taxonomic changes and images with appropriate [[Creative Commons]] licenses.

Many encyrtid genera are difficult to identify from photographs, but some are more recognizable. The photographs in the gallery below show some of the more recognizable ones. Please note that identifications of any chalcidoid wasps from photographs are difficult, and all should be treated with some doubt, unless the specimen was collected and details checked with a microscope.
&lt;gallery mode=packed heights=200&gt;
Anicetus iNat 180762161 d.jpg|''Anicetus'' spp. are endoparasitoids of wax scale and soft scale insects
Aenasius 2019 08 25 9691.jpg|''Aenasius'' sp.
Anagyrus inaturalist 205249827.jpg|''Anagyrus'' sp.
Apoleptomastix iNat 165108113.jpg|''Apoleptomastix'' sp.
Callipteroma sexguttata iNat 150291293.jpg|''Callipteroma sexguttata''
Cerchysius inaturalist 144151918.jpg|''Cerchysius'' sp.
Cheiloneurus inaturalist 327253478 04.jpg|''Cheiloneurus'' sp.
Cryptanusia iNat 19192414 magriet b.jpg|''Cryptanusia'' sp.
Diversinervus inat 196926650.jpg|''Diversinervus'' sp.
Homalotylus iNat 135531367.jpg|''Homalotylus'' sp.
Leptomastix inaturalist 256980040.jpg|''Leptomastix'' sp.
Microterys iN 253517641.jpg|''Microterys'' sp.
Neocladia inaturalist 316904882.jpg|''Neocladia'' sp.
Encyrtidae iNat 178385702 08.jpg|''Pentelicus'' sp.
Tachinaephagus iNat 148445032.jpg|''Tachinaephagus'' sp.

&lt;/gallery&gt;

== Encyrtidae for biological control of crop pests ==
Many encyrtid species are parasitoids of scale insects, some of which are pests that reduce productivity of agricultural crops across the world. Many scale insects have spread to countries where they have few natural enemies; in these cases, the introduction and spread of their wasp parasitoids can be used as a form of biological control to reduce their economic impact.&lt;ref name=Kapranas2015&gt;Kapranas, A., &amp; Tena, A. (2015). Encyrtid parasitoids of soft scale insects: Biology, behavior, and their use in biological control. Annual Review of Entomology, 60, 195-211. https://www.annualreviews.org/doi/10.1146/annurev-ento-010814-021053&lt;/ref&gt;&lt;ref name=Legner2024&gt;Legner, E. F., Economic gains &amp; analysis of success in biological pest control. University of California, Riverside. http://www.faculty.ucr.edu/~legneref/identify/museum5.htm (accessed 10 March 2024)&lt;/ref&gt;
&lt;br&gt;

==Links==
[http://African%20Arthropods/Afrotropical%20Encyrtidae%20Key Key to the genera of Afrotropical Encyrtidae]&lt;br&gt;
[https://www.waspweb.org/Chalcidoidea/Encyrtidae/index.htm WaspWeb: Encyrtid wasps of the Afrotropical Region]

==References==
{{reflist}}

[[Category:African Arthropods|Encyrtidae]]</text>
      <sha1>pbggi8jeyjvfbbmr2sh4eyzci7wm0o1</sha1>
    </revision>
    <revision>
      <id>2805781</id>
      <parentid>2805780</parentid>
      <timestamp>2026-04-21T14:25:30Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Encyrtidae - genera */</comment>
      <origin>2805781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4756" sha1="cmrznwg9w2z63jslel7m1r6yzwpa8g3" xml:space="preserve">There are more than 640 described species of Afrotropical Encyrtidae in about 130 genera. More than 380 of these species have been found in South Africa.

==Diagnostic features of Encyrtidae==
In macro photographs, encyrtids are often recognizable using these features:&lt;br&gt;
# Large mesopleuron, usually covering more than half of the thorax (mesosoma) in side view.
# Mid coxae join the thorax near the middle of the mesopleuron.
# Mesoscutum transverse (width greater than length), and generally not segmented by notauli.
# The axillae are usually visible as triangles with two long sides, one adjacent to the mesoscutum and the other adjacent to the scutellum; the short side of the triangle is adjacent to the base of the forewing. The axillae touch, or nearly touch, medially; appearing as wedges between the mesoscutum and the scutellum.
# Cercal plates (at the base of the cercal bristles) are advanced; rather than being near the posterior tip of the metasoma (abdomen); they are usually within the anterior (front) two-thirds of the metasoma.
#In Encyrtidae with fully developed wings, the marginal vein of the forewing is usually shorter than stigmal vein; there is also an oblique band on the forewing that lacks setae (a linea calva).
&lt;gallery mode=packed heights=300&gt;
Encyrtidae lateral view with annotations.jpg
Encyrtidae dorsal view with annotations.jpg
Encyrtidae dorso-lateral view with annotations.jpg
Forewing of Anagyrus minipedicellus from Malaysia Fig7 Zu et al 2018.jpg
&lt;/gallery&gt;

== Encyrtidae - genera ==
An illustrated key to the encyrtid genera of the Afrotropics was published in 1979.&lt;ref name=Prinsloo&amp;Anneke1979&gt;Prinsloo, G. L., &amp; Annecke, D. P. (1979). A key to the genera of Encyrtidae from the Ethiopian region, with descriptions of three new genera (Hymenoptera: Chalcidoidea). Journal of the Entomological Society of southern Africa, 42(2), 349-382. [https://journals.co.za/doi/pdf/10.10520/AJA00128789_2641 PDF]&lt;/ref&gt;. This key has been updated [[African Arthropods/Afrotropical Encyrtidae Key|'''here''']] to include some taxonomic changes and images with appropriate [[Creative Commons]] licenses.

Many encyrtid genera are difficult to identify from photographs, but some are more recognizable. The photographs in the gallery below show some of the more recognizable ones. Please note that identifications of any chalcidoid wasps from photographs are difficult, and all should be treated with some doubt, unless the specimen was collected and details checked with a microscope.
&lt;gallery mode=packed heights=200&gt;
Anicetus iNat 180762161 d.jpg|''[[w:Anicetus (wasp)|Anicetus]]'' spp. are endoparasitoids of wax scale and soft scale insects
Aenasius 2019 08 25 9691.jpg|''[[w:Aenasius|Aenasius]]'' spp. are parasitoids of mealybug nymphs
Anagyrus inaturalist 205249827.jpg|''Anagyrus'' sp.
Apoleptomastix iNat 165108113.jpg|''Apoleptomastix'' sp.
Callipteroma sexguttata iNat 150291293.jpg|''Callipteroma sexguttata''
Cerchysius inaturalist 144151918.jpg|''Cerchysius'' sp.
Cheiloneurus inaturalist 327253478 04.jpg|''Cheiloneurus'' sp.
Cryptanusia iNat 19192414 magriet b.jpg|''Cryptanusia'' sp.
Diversinervus inat 196926650.jpg|''Diversinervus'' sp.
Homalotylus iNat 135531367.jpg|''Homalotylus'' sp.
Leptomastix inaturalist 256980040.jpg|''Leptomastix'' sp.
Microterys iN 253517641.jpg|''Microterys'' sp.
Neocladia inaturalist 316904882.jpg|''Neocladia'' sp.
Encyrtidae iNat 178385702 08.jpg|''Pentelicus'' sp.
Tachinaephagus iNat 148445032.jpg|''Tachinaephagus'' sp.

&lt;/gallery&gt;

== Encyrtidae for biological control of crop pests ==
Many encyrtid species are parasitoids of scale insects, some of which are pests that reduce productivity of agricultural crops across the world. Many scale insects have spread to countries where they have few natural enemies; in these cases, the introduction and spread of their wasp parasitoids can be used as a form of biological control to reduce their economic impact.&lt;ref name=Kapranas2015&gt;Kapranas, A., &amp; Tena, A. (2015). Encyrtid parasitoids of soft scale insects: Biology, behavior, and their use in biological control. Annual Review of Entomology, 60, 195-211. https://www.annualreviews.org/doi/10.1146/annurev-ento-010814-021053&lt;/ref&gt;&lt;ref name=Legner2024&gt;Legner, E. F., Economic gains &amp; analysis of success in biological pest control. University of California, Riverside. http://www.faculty.ucr.edu/~legneref/identify/museum5.htm (accessed 10 March 2024)&lt;/ref&gt;
&lt;br&gt;

==Links==
[http://African%20Arthropods/Afrotropical%20Encyrtidae%20Key Key to the genera of Afrotropical Encyrtidae]&lt;br&gt;
[https://www.waspweb.org/Chalcidoidea/Encyrtidae/index.htm WaspWeb: Encyrtid wasps of the Afrotropical Region]

==References==
{{reflist}}

[[Category:African Arthropods|Encyrtidae]]</text>
      <sha1>cmrznwg9w2z63jslel7m1r6yzwpa8g3</sha1>
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  <page>
    <title>User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells</title>
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    <id>304065</id>
    <revision>
      <id>2805812</id>
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        <username>Dc.samizdat</username>
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      <comment>/* Introduction */</comment>
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      <text bytes="144141" sha1="gzhmtnts08jl9jgaxsegft87y37bzys" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter found the central section 8&lt;sub&gt;3&lt;/sub&gt; first, and gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways. Consequently the 120-cell can be constructed as an infinitesimally mobile rigid geodesic 3-sphere: a 4-dimensional tensegrity sphere. The 120-cell's 1200 edges need only be tension cables, provided that a disjoint 600 of the 120 5-cells' 1200 edges are included as compression struts, in parallel pairs.|name=tensegrity 120-cell}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
|-
![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
|-
|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}
Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, up to the heptagon {7}. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite web| url = https://www.youtube.com/watch?v=9sM44p385Ws| title = Vector Equilibrium | first = R. Buckminster | last = Fuller | author-link=W:Buckminster Fuller | year = 1975 | work= Everything I Know Sessions | place = Philadelphia}}
* {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=Bucky Fuller and the languages of geometry|title-link=User:Dc.samizdat#Bucky Fuller and the languages of geometry|journal=Wikiversity|ref={{SfnRef|Christie: On Fuller's use of language|2024}}}}
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* {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=Archimedean and Catalan solid hulls with their Weyl orbit definitions|title-link=Wikimedia:File:Archimedean and Catalan solid hulls with their Weyl orbit definitions.svg|journal=Wikimedia Commons|ref={{SfnRef|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}}}}
* {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin Geometric Group Theory|journal=PowerPoint|url=https://theoryofeverything.org/TOE/JGM/Quaternion%20Coxeter-Dynkin%20Geometric%20Group%20Theory-2b.pdf|ref={{SfnRef|Moxness: Quaternion graphics software|2023}}}}

=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
* {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}}

{{Refend}}</text>
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      <text bytes="144134" sha1="5hcym9qhkpwp6t91dhab4q98on3wloc" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter found the central section 8&lt;sub&gt;3&lt;/sub&gt; first, and gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways. Consequently the 120-cell can be constructed as an infinitesimally mobile rigid geodesic 3-sphere: a 4-dimensional tensegrity sphere. The 120-cell's 1200 edges need only be tension cables, provided that a disjoint 600 of the 120 5-cells' 1200 edges are included as compression struts, in parallel pairs.|name=tensegrity 120-cell}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
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|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
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![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
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|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
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Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, up to the heptagon {7}. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
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}}
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*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
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=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
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* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
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* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
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{{Refend}}</text>
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{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter found the central section 8&lt;sub&gt;3&lt;/sub&gt; first, and gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways. Consequently the 120-cell can be constructed as an infinitesimally mobile rigid geodesic 3-sphere: a 4-dimensional tensegrity sphere. The 120-cell's 1200 edges need only be tension cables, provided that a disjoint 600 of the 120 5-cells' 1200 edges are included as compression struts, in parallel pairs.|name=tensegrity 120-cell}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
|-
![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
|-
|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}
Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, up to the heptagon {7}. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
* {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}}

{{Refend}}</text>
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      <text bytes="143867" sha1="n367r44f3rrxpf14lhja19677bs42ro" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter found the central section 8&lt;sub&gt;3&lt;/sub&gt; first, and gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
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![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
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|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
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Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, up to the heptagon {7}. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
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{{Refend}}</text>
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      <text bytes="143885" sha1="dqpyi11s6in6zetles9v8xwk7yc7z0f" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter was the first to describe the central section 8&lt;sub&gt;3&lt;/sub&gt;, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
|-
![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
|-
|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}
Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, up to the heptagon {7}. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
* {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}}

{{Refend}}</text>
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      <text bytes="143918" sha1="t9nkpwpaaa7778zgaxfmd3xq6jjomfh" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter was the first to describe the central section 8&lt;sub&gt;3&lt;/sub&gt;, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
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![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
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|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
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Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every kind of regular 1-, 2-, 3-, and 4-polytope, except for the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
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}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
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{{Refend}}</text>
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      <text bytes="143906" sha1="0qhbvgljv8gyvajnzfn8e50sdg86xam" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter was the first to describe the central section 8&lt;sub&gt;3&lt;/sub&gt;, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
|-
![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
|-
|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}
Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
* {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}}

{{Refend}}</text>
      <sha1>0qhbvgljv8gyvajnzfn8e50sdg86xam</sha1>
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    <revision>
      <id>2805889</id>
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      <comment>/* How many building blocks, how many ways */</comment>
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      <text bytes="143908" sha1="0jj95ltggpef8gfrpsyixpzwbge0syp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|March 2024 - January 2026}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] &lt;sub&gt;5&lt;/sub&gt;{3,5,3}&lt;sub&gt;5&lt;/sub&gt;, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}&lt;sub&gt;5&lt;/sub&gt;, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box.

Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group &lt;math&gt;A_5&lt;/math&gt;, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share &lt;math&gt;A_5&lt;/math&gt; symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin &amp; Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles.

The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are.

== The real hemi-icosahedron ==

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The &quot;Combined hulls&quot; is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]]
We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}} The quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length &lt;math&gt;1&lt;/math&gt; in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt; &lt;/small&gt;(104.5°), not the 16-cell faces of edge length &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; (90°), not the 24-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt; (60°), and not the 600-cell faces of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}/\phi \approx 0.874&lt;/math&gt;&lt;/small&gt; (36°).|name=Moxness 60-point triangle faces}}

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]]
Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and  running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. 

[[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]]
But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length &lt;math&gt;1&lt;/math&gt;. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length &lt;math&gt;1&lt;/math&gt; is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length &lt;math&gt;1&lt;/math&gt; that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}}

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]]
Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8&lt;sub&gt;3&lt;/sub&gt;, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}} Coxeter was the first to describe the central section 8&lt;sub&gt;3&lt;/sub&gt;, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8&lt;sub&gt;3&lt;/sub&gt; section, he may have been the first person to ''see'' it.

The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4&lt;sub&gt;0&lt;/sub&gt; of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8&lt;sub&gt;3&lt;/sub&gt; of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells.

The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope.

Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1&lt;sub&gt;3&lt;/sub&gt;), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8&lt;sub&gt;3&lt;/sub&gt;), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''&lt;sub&gt;3&lt;/sub&gt; hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8&lt;sub&gt;3&lt;/sub&gt; has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1&lt;sub&gt;3&lt;/sub&gt; through 7&lt;sub&gt;3&lt;/sub&gt;, and 60 instances of the central section 8&lt;sub&gt;3&lt;/sub&gt;.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}}

[[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]]
We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra.

[[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.&lt;ref&gt;{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment &amp; Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}&lt;/ref&gt;]]
A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather &amp; Petrie|1938|p=4|ps=; &quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Eleven ==

Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. 

''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron.

{|class=&quot;wikitable floatright&quot; width=450
!colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position
|-
![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]]
![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}}
|- valign=top
|[[File:Regular_star_figure_6(5,2).svg|240px]]&lt;BR&gt;The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are visible. The other five edges, connecting the points of the six 5-cell pentagrams, are visible in the 6{5} projection below, the orthogonal view:&lt;BR&gt;[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the completely orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension.
|[[File:Regular_star_figure_2(15,4).svg|240px]]&lt;BR&gt;The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.&lt;BR&gt;[[File:Regular_star_figure_2(15,2).svg|240px]]
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![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]]
![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]]
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|colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
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Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways.

There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢&lt;sub&gt;4&lt;/sub&gt; the operation [15]𝑹&lt;sub&gt;q3,q3&lt;/sub&gt; is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt; the operation [1200]𝑹&lt;sub&gt;q3,q13&lt;/sub&gt; is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex  75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}}

The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. 

The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six.

== The eleventh chord ==
[[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]]
In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length &lt;math&gt;\phi^2&lt;/math&gt;, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: 

* 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11)
* 15.5° {30/1} + 120° (30/10) = 135.5° {30/11)

and its chord length is the linear sum of five shorter chords: 

* 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11)

Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the &lt;math&gt;\phi^2&lt;/math&gt; {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length &lt;small&gt;&lt;math&gt;\phi^{-2}&lt;/math&gt;&lt;/small&gt;. The 25.2° {30/2} pentagon face diagonal chord of length &lt;small&gt;&lt;math&gt;\phi^{-1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 41.1° 5-cell isocline chord of length &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 69.8° chord of length &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^3&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length &lt;small&gt;&lt;math&gt;\phi^2&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\phi^4&lt;/math&gt;&lt;/small&gt; times the {30/1} edge length.}}  

The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: 

* 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord
* 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord
* 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord

[[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]]
The last of those bisections trisects the {30/11} chord into three distinct shorter chords: 

* 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord

The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. 

Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. 

In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length &lt;small&gt;&lt;math&gt;\sqrt{3}/\phi&lt;/math&gt;&lt;/small&gt;, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram.

In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts.

Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways.

{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}}

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

=== Building the building blocks themselves ===

We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]].

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}}

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is the symmetry order of the 120-cell.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;H_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;5 \times 14400 / 5 = 14400&lt;/math&gt;&lt;/small&gt;.

=== Building with sticks ===

[[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]]
We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (&lt;math&gt;A_4&lt;/math&gt; or &lt;math&gt;B_4&lt;/math&gt;).

All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to the polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals &lt;math&gt;d_k&lt;/math&gt; of an &lt;math&gt;n&lt;/math&gt;-gon (with edge &lt;math&gt;d_0=1&lt;/math&gt;) as linear combinations of diagonals.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt;, and to the winding number or density of the polygram (its denominator &lt;math&gt;d&lt;/math&gt;).{{Sfn|Kappraff &amp; Adamson|2004}} The largest &lt;math&gt;k&lt;/math&gt; of any major chord in the 120-cell is 30, and the polygrams &lt;small&gt;&lt;math&gt;\{30/d\}&lt;/math&gt;&lt;/small&gt; represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes.

== Concentric 120-cells ==

The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end.

To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells.

That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells.

Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: 

* The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells.
* The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells.
* The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells.
In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut.

If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut.

[[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}.

The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;. The center section plus either of the smaller sections is &lt;small&gt;&lt;math&gt;\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;, the larger golden section. Each of the two smaller sections is &lt;small&gt;&lt;math&gt;\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.}}

The smaller golden sections &lt;small&gt;&lt;math&gt;\Phi \approx 0.618&lt;/math&gt;&lt;/small&gt; of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes.

Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell.

The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. 

=== Bitruncating the 5-cells ===
The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect.

[[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a  {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]]
The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle.

The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations.

[[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]]
The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure.

The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core (&quot;Admiral of the fleet&quot;) is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} 

We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S&lt;sup&gt;3&lt;/sup&gt;}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1.

In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}}

In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}}

The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint)  bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord).

The section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes.

A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle.

We find the 25.2° chord as the edge of the non-central section 6&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8&lt;sub&gt;3&lt;/sub&gt; (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;|ps=; &lt;math&gt;e_2e_3C_{120}\ RID\ P_5\ P_6\ tT&lt;/math&gt;}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells.

The 60 distinct section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces.

=== Rectifying the 16-cells ===
Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge.

Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks.

Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell.

=== Rectifying the 5-cells ===
In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius.

[[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]]
Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron.

The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}}

If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells,  240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell.

Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections.

The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge).

=== Truncating the 5-cells ===
[[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]]
A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's).

The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. 

When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}.

The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. 

The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords.

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a cubical shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry.  

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord.

We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra.

The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} 

Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the &lt;math&gt;H_4&lt;/math&gt; polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices.

The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell.

This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique &lt;math&gt;F_4&lt;/math&gt; symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction.

We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is &lt;math&gt;50/25 = 2/1&lt;/math&gt;, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is &lt;math&gt;600/225 = 8/3 &lt;/math&gt;.

We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points.

We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position.

We observe inscribed 5-cells only in the 120-cell. The 600-cell has &lt;math&gt;5^2 = 25&lt;/math&gt; distinct 24-cells inscribed in 120 vertices, and is a regular compound of &lt;math&gt;5&lt;/math&gt; disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has &lt;math&gt;5^2 \times 3^2 = 225&lt;/math&gt; distinct 24-cells inscribed in 600 vertices, and is a regular compound of &lt;math&gt;5^2 = 25&lt;/math&gt; disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells.

[[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]]
The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell.

The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes.

Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways).

The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart.

Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices.

If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract.

== Conclusion ==

Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Acknowledgements ==

...

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=Bucky Fuller and the languages of geometry|title-link=User:Dc.samizdat#Bucky Fuller and the languages of geometry|journal=Wikiversity|ref={{SfnRef|Christie: On Fuller's use of language|2024}}}}
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* {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin Geometric Group Theory|journal=PowerPoint|url=https://theoryofeverything.org/TOE/JGM/Quaternion%20Coxeter-Dynkin%20Geometric%20Group%20Theory-2b.pdf|ref={{SfnRef|Moxness: Quaternion graphics software|2023}}}}

=== 11-cell ===
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A &amp; M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin &amp; Lanier|2007}}}}
*{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin &amp; Hamlin|2007}}}}
*{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
=== Illustrations ===
* {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}}
* {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}}
* {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}}
* {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}}
* {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}}
* {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}}
* {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}}
* {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}}
* {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}}
* {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}}
* {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}}
* {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}}
* {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}}
* {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}}
* {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 24-cell|2024}}}}
* {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 600-cell|2024}}}}
* {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen &amp; Goucher et al. eds. 120-cell|2024}}}}
* {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}}
* {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}}

{{Refend}}</text>
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== Introduction ==
This page about ''Power series'' for the course [[Complex Analysis]] can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Power%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Power%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Power series'' are considered in detail:
* (1) Definition of power series,
* (2) Radius of convergence
* (3) Taylor series

==Definition==
Power series &lt;math display=&quot;inline&quot;&gt;
  p(x)
&lt;/math&gt; is in [[Calculus]] a [[w:en:series|series]] of the following form

:&lt;math display=&quot;block&quot;&gt;
  p(x) = \sum_{n=0}^\infty p_n (x-x_0)^n
&lt;/math&gt;
with
* any [[w:en:Sequence (Mathematics)|sequence]] &lt;math display=&quot;inline&quot;&gt;
  (p_n)_{n \in \mathbb N_0}
&lt;/math&gt; [[w:en:real number|real]] or [[w:en:complex number|complex number]]
* the 'center of series' is &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;.

==Reference to real analysis==
Potency series play an important role in the [[w:de:Complex Analysis|Complex Analysis]] and often allow a meaningful continuation [[w:de:Reelle Funktion|real function]] into the complex numerical level. In particular, the question arises for which real or complex numbers converge a potency series. This question leads to the term [[w:en:Radius of convergence|radius of convergence]].

==Convergence radius==
The largest number &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; is defined as the convergence radius of a potency series around the development point &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;, for which the potency series for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;
   r &gt; |x-x_0|
&lt;/math&gt; The [[w:en:open ball|open ball]]  &lt;math display=&quot;inline&quot;&gt;
  U_r(x_0)
&lt;/math&gt; with radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; around &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; are called 'convergence circle'. The convergence radius is therefore the radius of the convergence circle. If the series is converged for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt;, it is said that the convergence radius is infinite. Converged only for &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;, the convergence radius is 0, the power series can be called ''nowhere convergent''.

===Calculation Convergence radius - Cauchy-Hadamard===
For power series, the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; can be calculated with the 'formula of Cauchy-Hadamard'. It shall apply:

:&lt;math display=&quot;block&quot;&gt;
  r = \frac{1}{\limsup\limits_{n\rightarrow\infty}\ \sqrt[n]{|a_n|}}
&lt;/math&gt;
In this context, &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{0} := +\infty
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{\infty} := 0
&lt;/math&gt; are defined

===Calculation Convergence radius - non-threatening coefficients===
In many cases, the convergence radius can also be calculated in a simpler way in the case that the power series has only non-zero coefficients by the following limit:

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;
Non-zero coefficients are necessary that the limit fraction inside the limit exists.

==Examples==
Each [[w:en:Polynomial|Polynomial function]] can be classified as a power series, in which [[w:de:fast alle|almost all]] coefficients &lt;math display=&quot;inline&quot;&gt;
  a_n
&lt;/math&gt; are equal to 0. Important other examples are [[w:de:Taylorreihe|Taylor series]] and [[w:en:Maclaurin series|Maclaurin series]]. Functions which can be represented by a power series are also called [[w:de:analytische Funktion|analytical Function]]. Here again by way of example the potency series representation of some known functions:

===Exponential function===
[[w:de:Exponentialfunktion|Exponential function]] &lt;math display=&quot;inline&quot;&gt;
  exp:\mathbb{C} \to \mathbb{C}\setminus \{0\}
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  e^z = \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!} = \frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dotsb
&lt;/math&gt;
for all &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{C}
&lt;/math&gt;, i.e., the convergence radius is infinite.

===Sinus function/cosine===
[[w:de:Sinus|Sinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!}\mp\dotsb 
&lt;/math&gt;
[[w:de:Kosinus|Kosinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!}\mp\dotsb 
&lt;/math&gt;

===Convergence radius for sin, cos, exp===
The [[w:en:radius of convergence|radius of convergence]] is infinite both for the sine, cosine and for the exponential function. The potency series representation results directly from the exponential function with the [[w:de:Eulersche Formel|Euler's formula]].

=== Animation of Approximation with Taylor polynomial ===
The following animation shows the approximation of &lt;math&gt;cos(z)&lt;/math&gt; with Taylor polynomials with an increasing degree.

[[File:Taylor cos.gif|450px|center|Approximation of cos(z) with a Taylor sum as animation]]

===Logarithm===
[[w:de:Logarithmus|Logarithm function]]:

:&lt;math display=&quot;block&quot;&gt;
  \ln(1+z) = \sum_{k=1}^\infty (-1)^{k+1} \frac{z^k}{k}= z-\frac{z^2}{2} + \frac{z^3}{3} -\frac{z^4}{4}+ \dotsb
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  |z| &lt; 1
&lt;/math&gt;, i.e. The convergence radius is 1, for &lt;math display=&quot;inline&quot;&gt;
  z=1
&lt;/math&gt; the series is convergent, for &lt;math display=&quot;inline&quot;&gt;
  z=-1
&lt;/math&gt; divergent.

===Root===
[[w:de:Wurzel (Mathematik)|root function]]:

:&lt;math display=&quot;block&quot;&gt;
  \sqrt{1+x} = 1 + \frac{1}{2} x-\frac{1}{2\cdot4} x^2+\frac{1\cdot3}{2\cdot4\cdot6} x^3 \mp \dotsb 
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  -1 \leq x \leq 1
&lt;/math&gt;,
i.e., the convergence radius in &lt;math display=&quot;inline&quot;&gt;
  \mathbb{R}
&lt;/math&gt; is 1 and the series converged both for &lt;math display=&quot;inline&quot;&gt;
  x=1
&lt;/math&gt; and for &lt;math display=&quot;inline&quot;&gt;
  x=-1
&lt;/math&gt;.

==Characteristics==
The power series is important in the function theory because holomorphic functions can always be developed locally in potency rows.
The following topics are dealt with in the course.

===Continuity - Differentiability===
Power series  [[w:de:Normale Konvergenz|are normal convergent]] within their circle of  convergent. This directly follows that each function defined by a power series is continuous. Furthermore, it follows that there [[w:de:gleichmäßige Konvergenz|uniform convergence]] on compact subsets of the circle of convergence. This justifies the term-by-term differentiation and integration of a power series and shows that power series are infinitely differentiable.

===Absolute convergence===
[[w:de:absolute Konvergenz|Absolute convergence]]  exists within the circle of convergence. No general statement can be made about the behaviour of a potency series on the edge of the convergence circle, but in some cases the [[w:de:Abelscher Grenzwertsatz|Abel limit theorem]] allows to make a statement.

===Uniqueness of the power series representation===
The power series representation of a function around a development point is clearly determined (identity set for potency rows). In particular, for a given development point, Taylor development is the only possible potency series development.

==Operations with potency series==
Potency rows &lt;math display=&quot;inline&quot;&gt;
  p
&lt;/math&gt; can be recorded as vectors in a vector space &lt;math display=&quot;inline&quot;&gt;
  (\mathbb{C}[z],+,\cdot,\mathbb{C})
&lt;/math&gt;.

===Addition and scalar multiplication===
Are &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; by two potency rows

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;.

===Scale multiplication===
If &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; are due to two potency rows and &lt;math display=&quot;inline&quot;&gt;
  c
&lt;/math&gt; is a fixed complex number, then &lt;math display=&quot;inline&quot;&gt;
  f+g
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  cf
&lt;/math&gt; are considered to be at least

:&lt;math display=&quot;block&quot;&gt;
  f(x)+g(x) = \sum_{n=0}^\infty (a_n + b_n) (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  cf(x) = \sum_{n=0}^\infty (c a_n) (x-x_0)^n
&lt;/math&gt;

===Multiplication===
The product of two potency rows with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; is a potency row with a convergence radius which is at least &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;. Since there is absolute convergence within the convergence circle, the following applies after [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align}
f(x)g(x) &amp;= \left(\sum_{n=0}^\infty a_n (x-x_0)^n\right)\left(\sum_{n=0}^\infty b_n (x-x_0)^n\right)\\
&amp;= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-x_0)^{i+j}
 = \sum_{n=0}^\infty \left( \sum_{i=0}^n a_i b_{n-i}\right) (x-x_0)^n
\end{align}
&lt;/math&gt;
The sequence defined by &lt;math display=&quot;inline&quot;&gt;
  \textstyle c_n = \sum_{i=0}^n a_i b_{n-i}
&lt;/math&gt; &lt;math display=&quot;inline&quot;&gt;
  (c_n)
&lt;/math&gt; is called [[w:de:Faltung (Mathematik)#Diskrete Faltung|Faltung]] or convolution of the two sequences &lt;math display=&quot;inline&quot;&gt;
  (a_n)
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  (b_n)
&lt;/math&gt;.

===Chain===
There were &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; two potency series

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_1)^n. \mbox{.und }  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with positive convergence radii and property

:&lt;math display=&quot;block&quot;&gt;
  b_0 = g(x_0) = x_1
&lt;/math&gt;.
The linking &lt;math display=&quot;inline&quot;&gt;
  f\circ g
&lt;/math&gt; of both functions can then be developed locally again [[w:de:analytische Funktion|analytical Function]] and thus by &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; into a potency series:

:&lt;math display=&quot;block&quot;&gt;
  (f\circ g)(x) = \sum_{n=0}^\infty c_n (x-x_0)^n
&lt;/math&gt;

===Taylor series===
According to  [[w:de:Satz von Taylor|Taylor's theorem]]:

:&lt;math display=&quot;block&quot;&gt;
  c_n = \frac{(f\circ g)^{(n)}(x_0)}{n!}
&lt;/math&gt;
With the [[w:de:Formel von Faà di Bruno|Formel von Faà di Bruno]], this expression can now be indicated in a closed formula as a function of the given series coefficients, since:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
f^{(n)}(g(x_0))
&amp;= f^{(n)}(x_1) \\
&amp;= n!\cdot a_n \\
g^{(m)}(x_0)
&amp;= m!\cdot b_m
\end{align}

&lt;/math&gt;
[[w:de:Multiindex|Multiindex]] procedure is obtained:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
c_n
&amp;=\frac{(f\circ g)^{(n)}(x_0)}{n!} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{f^{(|\boldsymbol{k}|)}(g(x_0))}{\boldsymbol{k}!}
 \prod_{m=1\atop k_m\ge1}^n \left(\frac{g^{(m)}(x_0)}{m!}\right)^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{|\boldsymbol{k}|! \cdot a_{|\boldsymbol{k}|}}{\boldsymbol{k}!}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} {{|\boldsymbol{k}|} \choose \boldsymbol{k}} \, a_{|\boldsymbol{k}|}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m}
\end{align}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  {{|\boldsymbol{k}|} \choose \boldsymbol{k}}
&lt;/math&gt; of the [[w:de:Multinomialkoeffizient|Multinomial coeffizient]] is &lt;math display=&quot;inline&quot;&gt;
  \boldsymbol{k}
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  T_{n}=\left\{ \boldsymbol{k}\in\mathbb{N}_{0}^{n} \, \Big | \, \sum_{j=1}^{n}j\cdot k_{j}=n\right\}
&lt;/math&gt; is the amount of all partitions of &lt;math display=&quot;inline&quot;&gt;
  n
&lt;/math&gt; (cf.

===Differentiation and integration===
A potency series can be differentiated in the interior of its convergence circle and the [[w:de:Differentialrechnung|derivative]] is obtained by elemental differentiation:

:&lt;math display=&quot;block&quot;&gt;
  
f^\prime(x) = \sum_{n=1}^\infty a_n n \left( x-x_0 \right)^{n-1}= \sum_{n=0}^\infty a_{n+1} \left(n+1 \right) \left( x-x_0 \right)^{n}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; can be differentiated as often as desired and the following applies:

:&lt;math display=&quot;block&quot;&gt;
  
f^{(k)}(x) = \sum_{n=k}^\infty \frac{n!}{(n-k)!} a_n (x-x_0)^{n-k}
           = \sum_{n=0}^\infty \frac{(n+k)!}{n!} a_{n+k} (x-x_0)^n

&lt;/math&gt;
Analogously, a [[w:de:Stammfunktion|antiderivative]] is obtained by means of a link-wise integration of a potency series:

:&lt;math display=&quot;block&quot;&gt;
  
\int f(x)\,\text{d}x = \sum_{n=0}^\infty \frac{a_n \left( x-x_0 \right)^{n+1}} {n+1} + C = \sum_{n=1}^\infty \frac{a_{n-1} \left( x-x_0 \right)^{n}} {n} + C

&lt;/math&gt;
In both cases, the convergence radius is equal to that of the original row.

==Presentation of functions as potency series==
Often, a given function is interested in a potency series representation – in particular to answer the question whether the function [[w:de:Analytische Funktion|analytic]] is. There are some strategies to determine a potential series representation, the most common by the [[w:de:Taylorreihe|Taylor series]]. Here, however, the problem often arises that one needs a closed representation for the discharges, which is often difficult to determine. However, there are some lighter strategies for [[w:de:gebrochen rationale Funktion|fuctional rational functions]]. As an example the function

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}
&lt;/math&gt;
to be considered.

===By means of the geometric series===
By factoring the denominator and subsequent use of the formula for the sum of a [[w:de:Geometrische Reihe|geometrischen Reihe]], a representation of the function as a product of infinite rows is obtained:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{(1-z)(3-z)}=\frac{z^2}{3}\cdot \frac{1}{1-z} \cdot \frac{1}{1-\frac{z}{3}} =  
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
    = \frac{z^2}{3}\cdot \left(\sum_{n=0}^\infty z^n \right) \cdot \left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right)= \frac{1}{3}\left(\sum_{n=2}^\infty z^n \right)\left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right) 
&lt;/math&gt;

===Product of geometric rows===
Both rows are potency rows around the development point &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; and can therefore be multiplied in the above-mentioned manner. The same result also provides the [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\sum_{n=0}^\infty \left(  \underbrace{\sum_{k=0}^n a_k b_{n-k}}_{c_n} \right) \cdot z^n 
&lt;/math&gt;
Series (mathematics)
===Coefficients of individual series===
The following shall apply:

:&lt;math display=&quot;block&quot;&gt;
  a_k=
\begin{cases}
0 &amp; \text{ für } k\in\{0,1\} \\
1 &amp; \text{ sonst}
\end{cases}
&lt;/math&gt;
and

:&lt;math display=&quot;block&quot;&gt;
  b_k=\frac{1}{3^k}.
&lt;/math&gt;

===Cauchy product formula===
This follows by applying the formula for the partial sum of a [[w:de:Geometrische Reihe|geometric series]]

:&lt;math display=&quot;block&quot;&gt;
  c_n = \sum_{k=0}^n a_k b_{n-k}= \sum_{k=2}^n \left( \frac{1}{3}\right)^{n-k}=\frac{1}{3^{n-2}}\sum_{k=0}^{n-2} 3^k= -\frac{1-3^{n-1}}{2 \cdot 3^{n-2}}
&lt;/math&gt;
as a closed representation for the coefficient sequence of the potency series. Thus, the potency series representation of the function around the development point 0 is given by

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \sum_{n=2}^\infty \frac{3}{2} \cdot \left(1-\frac{1}{3^{n-1}} \right) \cdot z^n
&lt;/math&gt;.

===Application of geometric rows or coefficient comparison===
As an alternative to geometrical series, it is an alternative to [[w:de:Koeffizientenvergleich|coefficient comparison]] is an alternative: One assumes that a power series representation exists for &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \frac{z^2}{z^2-4z+3}= \sum_{n=0}^\infty c_n z^n
&lt;/math&gt;
The function &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; has the unknown coefficient sequence &lt;math display=&quot;inline&quot;&gt;
  (c_n)_{n \in \mathbb{N}}
&lt;/math&gt;. After multiplication of the denominator and an index shift, the identity results:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} z^2 &amp; = (z^2-4z+3)\sum_{n=0}^\infty c_n z^n \\&amp; = \sum_{n=2}^\infty c_{n-2} z^n - \sum_{n=1}^\infty 4c_{n-1} z^n + \sum_{n=0}^\infty 3c_n z^n \\&amp; = 3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n \end{align}
&lt;/math&gt;
The potency series &lt;math display=&quot;inline&quot;&gt;
  g(z):=z^2Series (mathematics)
&lt;/math&gt; is compared with the potency series &lt;math display=&quot;inline&quot;&gt;
  h(z):=3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n
&lt;/math&gt;. Both potency rows have the same development point &lt;math display=&quot;inline&quot;&gt;
  z_o=0
&lt;/math&gt;. Therefore, the coefficients of both potency rows must also correspond. Thus, the coefficient of (698-1047-1731592552598-341-99 must be &lt;math display=&quot;inline&quot;&gt;
  0=3c_0
&lt;/math&gt;, for which the coefficient of &lt;math display=&quot;inline&quot;&gt;
  z^1
&lt;/math&gt; applies &lt;math display=&quot;inline&quot;&gt;
  0=3c_1-4c_0
&lt;/math&gt;, ...

===Recursion formula for coefficients===
However, since two potency rows are exactly the same when their coefficient sequences correspond, the coefficient comparison results

:&lt;math display=&quot;block&quot;&gt;
  c_0=0,\ c_1=0,\ c_2=\frac{1}{3}
&lt;/math&gt;
and the recursion equation

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;;
the above closed representation follows from the complete induction.

===Benefits coefficient comparison===
The method by means of coefficient comparison also has the advantage that other development points than &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; are possible. Consider the development point &lt;math display=&quot;inline&quot;&gt;
  z_1=-1 
&lt;/math&gt; as an example. First, the broken rational function must be shown as a polynomial in &lt;math display=&quot;inline&quot;&gt;
  (z-z_1)=(z+1)
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=\frac{(z+1)^2-2(z+1)+1}{(z+1)^2-6(z+1)+8}
&lt;/math&gt;

===Other points of development===
Analogously to the top, it is now assumed that a formal potency series around the development point exists with unknown coefficient sequence and multiplied by the denominator:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} (z+1)^2-2(z+1)+1 &amp; = ((z+1)^2 - 6(z+1)+8)\sum_{n=0}^\infty c_n (z+1)^n \\ &amp; = 8c_0+(z+1)(8c_1-6c_0) + \\ 
   &amp;  \qquad +  \sum_{n=2}^\infty (c_{n-2}-6c_{n-1}+8c_n)(z+1)^n \end{align}
&lt;/math&gt;
Again, by means of coefficient comparison

:&lt;math display=&quot;block&quot;&gt;
  c_0=\frac{1}{8},\ c_1=-\frac{5}{32},\ c_2=-\frac{1}{128}
&lt;/math&gt;
and as a recursion equation for the coefficients:

:&lt;math display=&quot;block&quot;&gt;
  c_n=\frac{-c_{n-2}+6c_{n-1}}{8}
&lt;/math&gt;

===Partial fraction decomposition===
If the given function is first applied [[w:de:Polynomdivision|Polynomial division]] and then [[w:de:Partialbruchzerlegung|Partial fraction decomposition]], the representation is obtained

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=1+\frac{4z-3}{(z-1)(z-3)}= 1 + \frac{1}{2} \cdot \frac{1}{1-z} - \frac{3}{2} \cdot \frac{1}{1-\frac{z}{3}}
&lt;/math&gt;.
By inserting the geometric row, the following results:

:&lt;math display=&quot;block&quot;&gt;
  {f(z)=1+\frac{1}{2} \cdot \sum_{n=0}^\infty z^n - \frac{3}{2} \cdot \sum_{n=0}^\infty \frac{1}{3^n}z^n = 1+\sum_{n=0}^\infty \frac{3}{2} \cdot\left( 1- \frac{1}{3^{n-1}} \right) z^n}
&lt;/math&gt;
The first three sequence elements of the coefficient sequence are all zero, and the representation given here agrees with the upper one.

==Generalizations==
Potency rows can be defined not only for &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{R}
&lt;/math&gt;, but are also generalizable. Thus, for example, R B is the [[w:de:Matrixexponential|Matrix exponential]] and the [[w:de:Matrixlogarithmus|Matrix logarithm]] generalizations of potency rows in the area of the [[w:de:Matrix (Mathematik)|square of matrices]].
If in a row also potencies with negative integer exponents occur, one speaks of a [[w:de:Laurent-Reihe|Laurent-series]]. If the exponent is allowed to accept broken values Series (mathematics), it is a [[w:de:Puiseux-Reihe|Puiseux-series]].
[[w:de:Formale Potenzreihe|Formal power series]] are used, for example, as [[w:de:erzeugende Funktion|generating Function]]s in [[w:de:Kombinatorik|combinations]] and [[w:de:Wahrscheinlichkeitstheorie|probabilitytheoery]] (for example [[w:de:wahrscheinlichkeitserzeugende Funktion|probability generating functions]]). In the [[w:de:Algebra|Algebra]], formal power series are examined over general [[w:de:Kommutativer Ring|complex ring]].

== References ==
* Kurt Endl, Wolfgang Luh: ''Analysis II.'' Aula-Verlag 1973, 7th edition 1989, ISBN 3-89104-455-0, pp. 85–89, 99.
* E. D. Solomentsev: [http://eom.springer.de/P/p074240.htm ''Power series.''] In: ''[[w:en:Encyclopedia of Mathematics|Encyclopedia of Mathematics]]''
&lt;references/&gt;

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* [[w:de:Kurs:Funktionentheorie/Potenzreihe|Kurs:Funktionentheorie/Potenzreihe]] URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Potenzreihe
* Date: 11/14/2024
* [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Potenzreihe&quot; srclang=&quot;de&quot; date=&quot;11/14/2024&quot; time=&quot;14:57&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Potenzreihe]]
&lt;/noinclude&gt;
&lt;!-- &lt;noinclude&gt;[[en:Course:Function theory/Potence series]]&lt;/noinclude&gt; --&gt;
[[Category:Wiki2Reveal|Wiki2Reveal]]

[[Category:Analytic functions|analytic function]]
[[Category:Sequences and series|Sequences]]
[[Category:Wiki2Reveal]]</text>
      <sha1>3o31it6h24skni00dfavhxnz3v7e8h6</sha1>
    </revision>
    <revision>
      <id>2805778</id>
      <parentid>2805777</parentid>
      <timestamp>2026-04-21T14:15:27Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Calculation Convergence radius - non-threatening coefficients */</comment>
      <origin>2805778</origin>
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== Introduction ==
This page about ''Power series'' for the course [[Complex Analysis]] can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Power%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Power%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Power series'' are considered in detail:
* (1) Definition of power series,
* (2) Radius of convergence
* (3) Taylor series

==Definition==
Power series &lt;math display=&quot;inline&quot;&gt;
  p(x)
&lt;/math&gt; is in [[Calculus]] a [[w:en:series|series]] of the following form

:&lt;math display=&quot;block&quot;&gt;
  p(x) = \sum_{n=0}^\infty p_n (x-x_0)^n
&lt;/math&gt;
with
* any [[w:en:Sequence (Mathematics)|sequence]] &lt;math display=&quot;inline&quot;&gt;
  (p_n)_{n \in \mathbb N_0}
&lt;/math&gt; [[w:en:real number|real]] or [[w:en:complex number|complex number]]
* the 'center of series' is &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;.

==Reference to real analysis==
Potency series play an important role in the [[w:de:Complex Analysis|Complex Analysis]] and often allow a meaningful continuation [[w:de:Reelle Funktion|real function]] into the complex numerical level. In particular, the question arises for which real or complex numbers converge a potency series. This question leads to the term [[w:en:Radius of convergence|radius of convergence]].

==Convergence radius==
The largest number &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; is defined as the convergence radius of a potency series around the development point &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;, for which the potency series for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;
   r &gt; |x-x_0|
&lt;/math&gt; The [[w:en:open ball|open ball]]  &lt;math display=&quot;inline&quot;&gt;
  U_r(x_0)
&lt;/math&gt; with radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; around &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; are called 'convergence circle'. The convergence radius is therefore the radius of the convergence circle. If the series is converged for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt;, it is said that the convergence radius is infinite. Converged only for &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;, the convergence radius is 0, the power series can be called ''nowhere convergent''.

===Calculation Convergence radius - Cauchy-Hadamard===
For power series, the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; can be calculated with the 'formula of Cauchy-Hadamard'. It shall apply:

:&lt;math display=&quot;block&quot;&gt;
  r = \frac{1}{\limsup\limits_{n\rightarrow\infty}\ \sqrt[n]{|a_n|}}
&lt;/math&gt;
In this context, &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{0} := +\infty
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{\infty} := 0
&lt;/math&gt; are defined

===Calculation Convergence radius - non-threatening coefficients===
In many cases, the convergence radius can also be calculated in a simpler way in the case that the power series has only non-zero coefficients by the following limit:

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;
With non-zero coefficients the fraction inside the limit is defined.

==Examples==
Each [[w:en:Polynomial|Polynomial function]] can be classified as a power series, in which [[w:de:fast alle|almost all]] coefficients &lt;math display=&quot;inline&quot;&gt;
  a_n
&lt;/math&gt; are equal to 0. Important other examples are [[w:de:Taylorreihe|Taylor series]] and [[w:en:Maclaurin series|Maclaurin series]]. Functions which can be represented by a power series are also called [[w:de:analytische Funktion|analytical Function]]. Here again by way of example the potency series representation of some known functions:

===Exponential function===
[[w:de:Exponentialfunktion|Exponential function]] &lt;math display=&quot;inline&quot;&gt;
  exp:\mathbb{C} \to \mathbb{C}\setminus \{0\}
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  e^z = \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!} = \frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dotsb
&lt;/math&gt;
for all &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{C}
&lt;/math&gt;, i.e., the convergence radius is infinite.

===Sinus function/cosine===
[[w:de:Sinus|Sinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!}\mp\dotsb 
&lt;/math&gt;
[[w:de:Kosinus|Kosinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!}\mp\dotsb 
&lt;/math&gt;

===Convergence radius for sin, cos, exp===
The [[w:en:radius of convergence|radius of convergence]] is infinite both for the sine, cosine and for the exponential function. The potency series representation results directly from the exponential function with the [[w:de:Eulersche Formel|Euler's formula]].

=== Animation of Approximation with Taylor polynomial ===
The following animation shows the approximation of &lt;math&gt;cos(z)&lt;/math&gt; with Taylor polynomials with an increasing degree.

[[File:Taylor cos.gif|450px|center|Approximation of cos(z) with a Taylor sum as animation]]

===Logarithm===
[[w:de:Logarithmus|Logarithm function]]:

:&lt;math display=&quot;block&quot;&gt;
  \ln(1+z) = \sum_{k=1}^\infty (-1)^{k+1} \frac{z^k}{k}= z-\frac{z^2}{2} + \frac{z^3}{3} -\frac{z^4}{4}+ \dotsb
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  |z| &lt; 1
&lt;/math&gt;, i.e. The convergence radius is 1, for &lt;math display=&quot;inline&quot;&gt;
  z=1
&lt;/math&gt; the series is convergent, for &lt;math display=&quot;inline&quot;&gt;
  z=-1
&lt;/math&gt; divergent.

===Root===
[[w:de:Wurzel (Mathematik)|root function]]:

:&lt;math display=&quot;block&quot;&gt;
  \sqrt{1+x} = 1 + \frac{1}{2} x-\frac{1}{2\cdot4} x^2+\frac{1\cdot3}{2\cdot4\cdot6} x^3 \mp \dotsb 
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  -1 \leq x \leq 1
&lt;/math&gt;,
i.e., the convergence radius in &lt;math display=&quot;inline&quot;&gt;
  \mathbb{R}
&lt;/math&gt; is 1 and the series converged both for &lt;math display=&quot;inline&quot;&gt;
  x=1
&lt;/math&gt; and for &lt;math display=&quot;inline&quot;&gt;
  x=-1
&lt;/math&gt;.

==Characteristics==
The power series is important in the function theory because holomorphic functions can always be developed locally in potency rows.
The following topics are dealt with in the course.

===Continuity - Differentiability===
Power series  [[w:de:Normale Konvergenz|are normal convergent]] within their circle of  convergent. This directly follows that each function defined by a power series is continuous. Furthermore, it follows that there [[w:de:gleichmäßige Konvergenz|uniform convergence]] on compact subsets of the circle of convergence. This justifies the term-by-term differentiation and integration of a power series and shows that power series are infinitely differentiable.

===Absolute convergence===
[[w:de:absolute Konvergenz|Absolute convergence]]  exists within the circle of convergence. No general statement can be made about the behaviour of a potency series on the edge of the convergence circle, but in some cases the [[w:de:Abelscher Grenzwertsatz|Abel limit theorem]] allows to make a statement.

===Uniqueness of the power series representation===
The power series representation of a function around a development point is clearly determined (identity set for potency rows). In particular, for a given development point, Taylor development is the only possible potency series development.

==Operations with potency series==
Potency rows &lt;math display=&quot;inline&quot;&gt;
  p
&lt;/math&gt; can be recorded as vectors in a vector space &lt;math display=&quot;inline&quot;&gt;
  (\mathbb{C}[z],+,\cdot,\mathbb{C})
&lt;/math&gt;.

===Addition and scalar multiplication===
Are &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; by two potency rows

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;.

===Scale multiplication===
If &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; are due to two potency rows and &lt;math display=&quot;inline&quot;&gt;
  c
&lt;/math&gt; is a fixed complex number, then &lt;math display=&quot;inline&quot;&gt;
  f+g
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  cf
&lt;/math&gt; are considered to be at least

:&lt;math display=&quot;block&quot;&gt;
  f(x)+g(x) = \sum_{n=0}^\infty (a_n + b_n) (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  cf(x) = \sum_{n=0}^\infty (c a_n) (x-x_0)^n
&lt;/math&gt;

===Multiplication===
The product of two potency rows with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; is a potency row with a convergence radius which is at least &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;. Since there is absolute convergence within the convergence circle, the following applies after [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align}
f(x)g(x) &amp;= \left(\sum_{n=0}^\infty a_n (x-x_0)^n\right)\left(\sum_{n=0}^\infty b_n (x-x_0)^n\right)\\
&amp;= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-x_0)^{i+j}
 = \sum_{n=0}^\infty \left( \sum_{i=0}^n a_i b_{n-i}\right) (x-x_0)^n
\end{align}
&lt;/math&gt;
The sequence defined by &lt;math display=&quot;inline&quot;&gt;
  \textstyle c_n = \sum_{i=0}^n a_i b_{n-i}
&lt;/math&gt; &lt;math display=&quot;inline&quot;&gt;
  (c_n)
&lt;/math&gt; is called [[w:de:Faltung (Mathematik)#Diskrete Faltung|Faltung]] or convolution of the two sequences &lt;math display=&quot;inline&quot;&gt;
  (a_n)
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  (b_n)
&lt;/math&gt;.

===Chain===
There were &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; two potency series

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_1)^n. \mbox{.und }  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with positive convergence radii and property

:&lt;math display=&quot;block&quot;&gt;
  b_0 = g(x_0) = x_1
&lt;/math&gt;.
The linking &lt;math display=&quot;inline&quot;&gt;
  f\circ g
&lt;/math&gt; of both functions can then be developed locally again [[w:de:analytische Funktion|analytical Function]] and thus by &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; into a potency series:

:&lt;math display=&quot;block&quot;&gt;
  (f\circ g)(x) = \sum_{n=0}^\infty c_n (x-x_0)^n
&lt;/math&gt;

===Taylor series===
According to  [[w:de:Satz von Taylor|Taylor's theorem]]:

:&lt;math display=&quot;block&quot;&gt;
  c_n = \frac{(f\circ g)^{(n)}(x_0)}{n!}
&lt;/math&gt;
With the [[w:de:Formel von Faà di Bruno|Formel von Faà di Bruno]], this expression can now be indicated in a closed formula as a function of the given series coefficients, since:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
f^{(n)}(g(x_0))
&amp;= f^{(n)}(x_1) \\
&amp;= n!\cdot a_n \\
g^{(m)}(x_0)
&amp;= m!\cdot b_m
\end{align}

&lt;/math&gt;
[[w:de:Multiindex|Multiindex]] procedure is obtained:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
c_n
&amp;=\frac{(f\circ g)^{(n)}(x_0)}{n!} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{f^{(|\boldsymbol{k}|)}(g(x_0))}{\boldsymbol{k}!}
 \prod_{m=1\atop k_m\ge1}^n \left(\frac{g^{(m)}(x_0)}{m!}\right)^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{|\boldsymbol{k}|! \cdot a_{|\boldsymbol{k}|}}{\boldsymbol{k}!}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} {{|\boldsymbol{k}|} \choose \boldsymbol{k}} \, a_{|\boldsymbol{k}|}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m}
\end{align}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  {{|\boldsymbol{k}|} \choose \boldsymbol{k}}
&lt;/math&gt; of the [[w:de:Multinomialkoeffizient|Multinomial coeffizient]] is &lt;math display=&quot;inline&quot;&gt;
  \boldsymbol{k}
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  T_{n}=\left\{ \boldsymbol{k}\in\mathbb{N}_{0}^{n} \, \Big | \, \sum_{j=1}^{n}j\cdot k_{j}=n\right\}
&lt;/math&gt; is the amount of all partitions of &lt;math display=&quot;inline&quot;&gt;
  n
&lt;/math&gt; (cf.

===Differentiation and integration===
A potency series can be differentiated in the interior of its convergence circle and the [[w:de:Differentialrechnung|derivative]] is obtained by elemental differentiation:

:&lt;math display=&quot;block&quot;&gt;
  
f^\prime(x) = \sum_{n=1}^\infty a_n n \left( x-x_0 \right)^{n-1}= \sum_{n=0}^\infty a_{n+1} \left(n+1 \right) \left( x-x_0 \right)^{n}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; can be differentiated as often as desired and the following applies:

:&lt;math display=&quot;block&quot;&gt;
  
f^{(k)}(x) = \sum_{n=k}^\infty \frac{n!}{(n-k)!} a_n (x-x_0)^{n-k}
           = \sum_{n=0}^\infty \frac{(n+k)!}{n!} a_{n+k} (x-x_0)^n

&lt;/math&gt;
Analogously, a [[w:de:Stammfunktion|antiderivative]] is obtained by means of a link-wise integration of a potency series:

:&lt;math display=&quot;block&quot;&gt;
  
\int f(x)\,\text{d}x = \sum_{n=0}^\infty \frac{a_n \left( x-x_0 \right)^{n+1}} {n+1} + C = \sum_{n=1}^\infty \frac{a_{n-1} \left( x-x_0 \right)^{n}} {n} + C

&lt;/math&gt;
In both cases, the convergence radius is equal to that of the original row.

==Presentation of functions as potency series==
Often, a given function is interested in a potency series representation – in particular to answer the question whether the function [[w:de:Analytische Funktion|analytic]] is. There are some strategies to determine a potential series representation, the most common by the [[w:de:Taylorreihe|Taylor series]]. Here, however, the problem often arises that one needs a closed representation for the discharges, which is often difficult to determine. However, there are some lighter strategies for [[w:de:gebrochen rationale Funktion|fuctional rational functions]]. As an example the function

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}
&lt;/math&gt;
to be considered.

===By means of the geometric series===
By factoring the denominator and subsequent use of the formula for the sum of a [[w:de:Geometrische Reihe|geometrischen Reihe]], a representation of the function as a product of infinite rows is obtained:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{(1-z)(3-z)}=\frac{z^2}{3}\cdot \frac{1}{1-z} \cdot \frac{1}{1-\frac{z}{3}} =  
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
    = \frac{z^2}{3}\cdot \left(\sum_{n=0}^\infty z^n \right) \cdot \left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right)= \frac{1}{3}\left(\sum_{n=2}^\infty z^n \right)\left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right) 
&lt;/math&gt;

===Product of geometric rows===
Both rows are potency rows around the development point &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; and can therefore be multiplied in the above-mentioned manner. The same result also provides the [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\sum_{n=0}^\infty \left(  \underbrace{\sum_{k=0}^n a_k b_{n-k}}_{c_n} \right) \cdot z^n 
&lt;/math&gt;
Series (mathematics)
===Coefficients of individual series===
The following shall apply:

:&lt;math display=&quot;block&quot;&gt;
  a_k=
\begin{cases}
0 &amp; \text{ für } k\in\{0,1\} \\
1 &amp; \text{ sonst}
\end{cases}
&lt;/math&gt;
and

:&lt;math display=&quot;block&quot;&gt;
  b_k=\frac{1}{3^k}.
&lt;/math&gt;

===Cauchy product formula===
This follows by applying the formula for the partial sum of a [[w:de:Geometrische Reihe|geometric series]]

:&lt;math display=&quot;block&quot;&gt;
  c_n = \sum_{k=0}^n a_k b_{n-k}= \sum_{k=2}^n \left( \frac{1}{3}\right)^{n-k}=\frac{1}{3^{n-2}}\sum_{k=0}^{n-2} 3^k= -\frac{1-3^{n-1}}{2 \cdot 3^{n-2}}
&lt;/math&gt;
as a closed representation for the coefficient sequence of the potency series. Thus, the potency series representation of the function around the development point 0 is given by

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \sum_{n=2}^\infty \frac{3}{2} \cdot \left(1-\frac{1}{3^{n-1}} \right) \cdot z^n
&lt;/math&gt;.

===Application of geometric rows or coefficient comparison===
As an alternative to geometrical series, it is an alternative to [[w:de:Koeffizientenvergleich|coefficient comparison]] is an alternative: One assumes that a power series representation exists for &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \frac{z^2}{z^2-4z+3}= \sum_{n=0}^\infty c_n z^n
&lt;/math&gt;
The function &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; has the unknown coefficient sequence &lt;math display=&quot;inline&quot;&gt;
  (c_n)_{n \in \mathbb{N}}
&lt;/math&gt;. After multiplication of the denominator and an index shift, the identity results:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} z^2 &amp; = (z^2-4z+3)\sum_{n=0}^\infty c_n z^n \\&amp; = \sum_{n=2}^\infty c_{n-2} z^n - \sum_{n=1}^\infty 4c_{n-1} z^n + \sum_{n=0}^\infty 3c_n z^n \\&amp; = 3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n \end{align}
&lt;/math&gt;
The potency series &lt;math display=&quot;inline&quot;&gt;
  g(z):=z^2Series (mathematics)
&lt;/math&gt; is compared with the potency series &lt;math display=&quot;inline&quot;&gt;
  h(z):=3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n
&lt;/math&gt;. Both potency rows have the same development point &lt;math display=&quot;inline&quot;&gt;
  z_o=0
&lt;/math&gt;. Therefore, the coefficients of both potency rows must also correspond. Thus, the coefficient of (698-1047-1731592552598-341-99 must be &lt;math display=&quot;inline&quot;&gt;
  0=3c_0
&lt;/math&gt;, for which the coefficient of &lt;math display=&quot;inline&quot;&gt;
  z^1
&lt;/math&gt; applies &lt;math display=&quot;inline&quot;&gt;
  0=3c_1-4c_0
&lt;/math&gt;, ...

===Recursion formula for coefficients===
However, since two potency rows are exactly the same when their coefficient sequences correspond, the coefficient comparison results

:&lt;math display=&quot;block&quot;&gt;
  c_0=0,\ c_1=0,\ c_2=\frac{1}{3}
&lt;/math&gt;
and the recursion equation

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;;
the above closed representation follows from the complete induction.

===Benefits coefficient comparison===
The method by means of coefficient comparison also has the advantage that other development points than &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; are possible. Consider the development point &lt;math display=&quot;inline&quot;&gt;
  z_1=-1 
&lt;/math&gt; as an example. First, the broken rational function must be shown as a polynomial in &lt;math display=&quot;inline&quot;&gt;
  (z-z_1)=(z+1)
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=\frac{(z+1)^2-2(z+1)+1}{(z+1)^2-6(z+1)+8}
&lt;/math&gt;

===Other points of development===
Analogously to the top, it is now assumed that a formal potency series around the development point exists with unknown coefficient sequence and multiplied by the denominator:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} (z+1)^2-2(z+1)+1 &amp; = ((z+1)^2 - 6(z+1)+8)\sum_{n=0}^\infty c_n (z+1)^n \\ &amp; = 8c_0+(z+1)(8c_1-6c_0) + \\ 
   &amp;  \qquad +  \sum_{n=2}^\infty (c_{n-2}-6c_{n-1}+8c_n)(z+1)^n \end{align}
&lt;/math&gt;
Again, by means of coefficient comparison

:&lt;math display=&quot;block&quot;&gt;
  c_0=\frac{1}{8},\ c_1=-\frac{5}{32},\ c_2=-\frac{1}{128}
&lt;/math&gt;
and as a recursion equation for the coefficients:

:&lt;math display=&quot;block&quot;&gt;
  c_n=\frac{-c_{n-2}+6c_{n-1}}{8}
&lt;/math&gt;

===Partial fraction decomposition===
If the given function is first applied [[w:de:Polynomdivision|Polynomial division]] and then [[w:de:Partialbruchzerlegung|Partial fraction decomposition]], the representation is obtained

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=1+\frac{4z-3}{(z-1)(z-3)}= 1 + \frac{1}{2} \cdot \frac{1}{1-z} - \frac{3}{2} \cdot \frac{1}{1-\frac{z}{3}}
&lt;/math&gt;.
By inserting the geometric row, the following results:

:&lt;math display=&quot;block&quot;&gt;
  {f(z)=1+\frac{1}{2} \cdot \sum_{n=0}^\infty z^n - \frac{3}{2} \cdot \sum_{n=0}^\infty \frac{1}{3^n}z^n = 1+\sum_{n=0}^\infty \frac{3}{2} \cdot\left( 1- \frac{1}{3^{n-1}} \right) z^n}
&lt;/math&gt;
The first three sequence elements of the coefficient sequence are all zero, and the representation given here agrees with the upper one.

==Generalizations==
Potency rows can be defined not only for &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{R}
&lt;/math&gt;, but are also generalizable. Thus, for example, R B is the [[w:de:Matrixexponential|Matrix exponential]] and the [[w:de:Matrixlogarithmus|Matrix logarithm]] generalizations of potency rows in the area of the [[w:de:Matrix (Mathematik)|square of matrices]].
If in a row also potencies with negative integer exponents occur, one speaks of a [[w:de:Laurent-Reihe|Laurent-series]]. If the exponent is allowed to accept broken values Series (mathematics), it is a [[w:de:Puiseux-Reihe|Puiseux-series]].
[[w:de:Formale Potenzreihe|Formal power series]] are used, for example, as [[w:de:erzeugende Funktion|generating Function]]s in [[w:de:Kombinatorik|combinations]] and [[w:de:Wahrscheinlichkeitstheorie|probabilitytheoery]] (for example [[w:de:wahrscheinlichkeitserzeugende Funktion|probability generating functions]]). In the [[w:de:Algebra|Algebra]], formal power series are examined over general [[w:de:Kommutativer Ring|complex ring]].

== References ==
* Kurt Endl, Wolfgang Luh: ''Analysis II.'' Aula-Verlag 1973, 7th edition 1989, ISBN 3-89104-455-0, pp. 85–89, 99.
* E. D. Solomentsev: [http://eom.springer.de/P/p074240.htm ''Power series.''] In: ''[[w:en:Encyclopedia of Mathematics|Encyclopedia of Mathematics]]''
&lt;references/&gt;

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* [[w:de:Kurs:Funktionentheorie/Potenzreihe|Kurs:Funktionentheorie/Potenzreihe]] URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Potenzreihe
* Date: 11/14/2024
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&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Potenzreihe&quot; srclang=&quot;de&quot; date=&quot;11/14/2024&quot; time=&quot;14:57&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Potenzreihe]]
&lt;/noinclude&gt;
&lt;!-- &lt;noinclude&gt;[[en:Course:Function theory/Potence series]]&lt;/noinclude&gt; --&gt;
[[Category:Wiki2Reveal|Wiki2Reveal]]

[[Category:Analytic functions|analytic function]]
[[Category:Sequences and series|Sequences]]
[[Category:Wiki2Reveal]]</text>
      <sha1>bfvtmbv1fl025kxb2fytdc45o83w707</sha1>
    </revision>
    <revision>
      <id>2805779</id>
      <parentid>2805778</parentid>
      <timestamp>2026-04-21T14:16:05Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Calculation Convergence radius - non-threatening coefficients */</comment>
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== Introduction ==
This page about ''Power series'' for the course [[Complex Analysis]] can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Power%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Power%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Power series'' are considered in detail:
* (1) Definition of power series,
* (2) Radius of convergence
* (3) Taylor series

==Definition==
Power series &lt;math display=&quot;inline&quot;&gt;
  p(x)
&lt;/math&gt; is in [[Calculus]] a [[w:en:series|series]] of the following form

:&lt;math display=&quot;block&quot;&gt;
  p(x) = \sum_{n=0}^\infty p_n (x-x_0)^n
&lt;/math&gt;
with
* any [[w:en:Sequence (Mathematics)|sequence]] &lt;math display=&quot;inline&quot;&gt;
  (p_n)_{n \in \mathbb N_0}
&lt;/math&gt; [[w:en:real number|real]] or [[w:en:complex number|complex number]]
* the 'center of series' is &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;.

==Reference to real analysis==
Potency series play an important role in the [[w:de:Complex Analysis|Complex Analysis]] and often allow a meaningful continuation [[w:de:Reelle Funktion|real function]] into the complex numerical level. In particular, the question arises for which real or complex numbers converge a potency series. This question leads to the term [[w:en:Radius of convergence|radius of convergence]].

==Convergence radius==
The largest number &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; is defined as the convergence radius of a potency series around the development point &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;, for which the potency series for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt; with &lt;math display=&quot;inline&quot;&gt;
   r &gt; |x-x_0|
&lt;/math&gt; The [[w:en:open ball|open ball]]  &lt;math display=&quot;inline&quot;&gt;
  U_r(x_0)
&lt;/math&gt; with radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; around &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; are called 'convergence circle'. The convergence radius is therefore the radius of the convergence circle. If the series is converged for all &lt;math display=&quot;inline&quot;&gt;
  x
&lt;/math&gt;, it is said that the convergence radius is infinite. Converged only for &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt;, the convergence radius is 0, the power series can be called ''nowhere convergent''.

===Calculation Convergence radius - Cauchy-Hadamard===
For power series, the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; can be calculated with the 'formula of Cauchy-Hadamard'. It shall apply:

:&lt;math display=&quot;block&quot;&gt;
  r = \frac{1}{\limsup\limits_{n\rightarrow\infty}\ \sqrt[n]{|a_n|}}
&lt;/math&gt;
In this context, &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{0} := +\infty
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  \tfrac{1}{\infty} := 0
&lt;/math&gt; are defined

===Calculation Convergence radius - non-zero coefficients===
In many cases, the convergence radius can also be calculated in a simpler way in the case that the power series has only non-zero coefficients by the following limit:

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;
With non-zero coefficients the fraction inside the limit is defined.

==Examples==
Each [[w:en:Polynomial|Polynomial function]] can be classified as a power series, in which [[w:de:fast alle|almost all]] coefficients &lt;math display=&quot;inline&quot;&gt;
  a_n
&lt;/math&gt; are equal to 0. Important other examples are [[w:de:Taylorreihe|Taylor series]] and [[w:en:Maclaurin series|Maclaurin series]]. Functions which can be represented by a power series are also called [[w:de:analytische Funktion|analytical Function]]. Here again by way of example the potency series representation of some known functions:

===Exponential function===
[[w:de:Exponentialfunktion|Exponential function]] &lt;math display=&quot;inline&quot;&gt;
  exp:\mathbb{C} \to \mathbb{C}\setminus \{0\}
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  e^z = \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!} = \frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dotsb
&lt;/math&gt;
for all &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{C}
&lt;/math&gt;, i.e., the convergence radius is infinite.

===Sinus function/cosine===
[[w:de:Sinus|Sinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!}\mp\dotsb 
&lt;/math&gt;
[[w:de:Kosinus|Kosinus]]:

:&lt;math display=&quot;block&quot;&gt;
  \cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!}\mp\dotsb 
&lt;/math&gt;

===Convergence radius for sin, cos, exp===
The [[w:en:radius of convergence|radius of convergence]] is infinite both for the sine, cosine and for the exponential function. The potency series representation results directly from the exponential function with the [[w:de:Eulersche Formel|Euler's formula]].

=== Animation of Approximation with Taylor polynomial ===
The following animation shows the approximation of &lt;math&gt;cos(z)&lt;/math&gt; with Taylor polynomials with an increasing degree.

[[File:Taylor cos.gif|450px|center|Approximation of cos(z) with a Taylor sum as animation]]

===Logarithm===
[[w:de:Logarithmus|Logarithm function]]:

:&lt;math display=&quot;block&quot;&gt;
  \ln(1+z) = \sum_{k=1}^\infty (-1)^{k+1} \frac{z^k}{k}= z-\frac{z^2}{2} + \frac{z^3}{3} -\frac{z^4}{4}+ \dotsb
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  |z| &lt; 1
&lt;/math&gt;, i.e. The convergence radius is 1, for &lt;math display=&quot;inline&quot;&gt;
  z=1
&lt;/math&gt; the series is convergent, for &lt;math display=&quot;inline&quot;&gt;
  z=-1
&lt;/math&gt; divergent.

===Root===
[[w:de:Wurzel (Mathematik)|root function]]:

:&lt;math display=&quot;block&quot;&gt;
  \sqrt{1+x} = 1 + \frac{1}{2} x-\frac{1}{2\cdot4} x^2+\frac{1\cdot3}{2\cdot4\cdot6} x^3 \mp \dotsb 
&lt;/math&gt;
for &lt;math display=&quot;inline&quot;&gt;
  -1 \leq x \leq 1
&lt;/math&gt;,
i.e., the convergence radius in &lt;math display=&quot;inline&quot;&gt;
  \mathbb{R}
&lt;/math&gt; is 1 and the series converged both for &lt;math display=&quot;inline&quot;&gt;
  x=1
&lt;/math&gt; and for &lt;math display=&quot;inline&quot;&gt;
  x=-1
&lt;/math&gt;.

==Characteristics==
The power series is important in the function theory because holomorphic functions can always be developed locally in potency rows.
The following topics are dealt with in the course.

===Continuity - Differentiability===
Power series  [[w:de:Normale Konvergenz|are normal convergent]] within their circle of  convergent. This directly follows that each function defined by a power series is continuous. Furthermore, it follows that there [[w:de:gleichmäßige Konvergenz|uniform convergence]] on compact subsets of the circle of convergence. This justifies the term-by-term differentiation and integration of a power series and shows that power series are infinitely differentiable.

===Absolute convergence===
[[w:de:absolute Konvergenz|Absolute convergence]]  exists within the circle of convergence. No general statement can be made about the behaviour of a potency series on the edge of the convergence circle, but in some cases the [[w:de:Abelscher Grenzwertsatz|Abel limit theorem]] allows to make a statement.

===Uniqueness of the power series representation===
The power series representation of a function around a development point is clearly determined (identity set for potency rows). In particular, for a given development point, Taylor development is the only possible potency series development.

==Operations with potency series==
Potency rows &lt;math display=&quot;inline&quot;&gt;
  p
&lt;/math&gt; can be recorded as vectors in a vector space &lt;math display=&quot;inline&quot;&gt;
  (\mathbb{C}[z],+,\cdot,\mathbb{C})
&lt;/math&gt;.

===Addition and scalar multiplication===
Are &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; by two potency rows

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;.

===Scale multiplication===
If &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; are due to two potency rows and &lt;math display=&quot;inline&quot;&gt;
  c
&lt;/math&gt; is a fixed complex number, then &lt;math display=&quot;inline&quot;&gt;
  f+g
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  cf
&lt;/math&gt; are considered to be at least

:&lt;math display=&quot;block&quot;&gt;
  f(x)+g(x) = \sum_{n=0}^\infty (a_n + b_n) (x-x_0)^n
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
  cf(x) = \sum_{n=0}^\infty (c a_n) (x-x_0)^n
&lt;/math&gt;

===Multiplication===
The product of two potency rows with the convergence radius &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt; is a potency row with a convergence radius which is at least &lt;math display=&quot;inline&quot;&gt;
  r
&lt;/math&gt;. Since there is absolute convergence within the convergence circle, the following applies after [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align}
f(x)g(x) &amp;= \left(\sum_{n=0}^\infty a_n (x-x_0)^n\right)\left(\sum_{n=0}^\infty b_n (x-x_0)^n\right)\\
&amp;= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-x_0)^{i+j}
 = \sum_{n=0}^\infty \left( \sum_{i=0}^n a_i b_{n-i}\right) (x-x_0)^n
\end{align}
&lt;/math&gt;
The sequence defined by &lt;math display=&quot;inline&quot;&gt;
  \textstyle c_n = \sum_{i=0}^n a_i b_{n-i}
&lt;/math&gt; &lt;math display=&quot;inline&quot;&gt;
  (c_n)
&lt;/math&gt; is called [[w:de:Faltung (Mathematik)#Diskrete Faltung|Faltung]] or convolution of the two sequences &lt;math display=&quot;inline&quot;&gt;
  (a_n)
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  (b_n)
&lt;/math&gt;.

===Chain===
There were &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  g
&lt;/math&gt; two potency series

:&lt;math display=&quot;block&quot;&gt;
  f(x) = \sum_{n=0}^\infty a_n (x-x_1)^n. \mbox{.und }  g(x) = \sum_{n=0}^\infty b_n (x-x_0)^n
&lt;/math&gt;
with positive convergence radii and property

:&lt;math display=&quot;block&quot;&gt;
  b_0 = g(x_0) = x_1
&lt;/math&gt;.
The linking &lt;math display=&quot;inline&quot;&gt;
  f\circ g
&lt;/math&gt; of both functions can then be developed locally again [[w:de:analytische Funktion|analytical Function]] and thus by &lt;math display=&quot;inline&quot;&gt;
  x_0
&lt;/math&gt; into a potency series:

:&lt;math display=&quot;block&quot;&gt;
  (f\circ g)(x) = \sum_{n=0}^\infty c_n (x-x_0)^n
&lt;/math&gt;

===Taylor series===
According to  [[w:de:Satz von Taylor|Taylor's theorem]]:

:&lt;math display=&quot;block&quot;&gt;
  c_n = \frac{(f\circ g)^{(n)}(x_0)}{n!}
&lt;/math&gt;
With the [[w:de:Formel von Faà di Bruno|Formel von Faà di Bruno]], this expression can now be indicated in a closed formula as a function of the given series coefficients, since:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
f^{(n)}(g(x_0))
&amp;= f^{(n)}(x_1) \\
&amp;= n!\cdot a_n \\
g^{(m)}(x_0)
&amp;= m!\cdot b_m
\end{align}

&lt;/math&gt;
[[w:de:Multiindex|Multiindex]] procedure is obtained:

:&lt;math display=&quot;block&quot;&gt;
  
\begin{align}
c_n
&amp;=\frac{(f\circ g)^{(n)}(x_0)}{n!} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{f^{(|\boldsymbol{k}|)}(g(x_0))}{\boldsymbol{k}!}
 \prod_{m=1\atop k_m\ge1}^n \left(\frac{g^{(m)}(x_0)}{m!}\right)^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} \frac{|\boldsymbol{k}|! \cdot a_{|\boldsymbol{k}|}}{\boldsymbol{k}!}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m} \\
&amp;=\sum_{\boldsymbol{k}\in T_n} {{|\boldsymbol{k}|} \choose \boldsymbol{k}} \, a_{|\boldsymbol{k}|}
 \prod_{ m=1\atop k_m\ge1}^n b_m^{k_m}
\end{align}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  {{|\boldsymbol{k}|} \choose \boldsymbol{k}}
&lt;/math&gt; of the [[w:de:Multinomialkoeffizient|Multinomial coeffizient]] is &lt;math display=&quot;inline&quot;&gt;
  \boldsymbol{k}
&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;
  T_{n}=\left\{ \boldsymbol{k}\in\mathbb{N}_{0}^{n} \, \Big | \, \sum_{j=1}^{n}j\cdot k_{j}=n\right\}
&lt;/math&gt; is the amount of all partitions of &lt;math display=&quot;inline&quot;&gt;
  n
&lt;/math&gt; (cf.

===Differentiation and integration===
A potency series can be differentiated in the interior of its convergence circle and the [[w:de:Differentialrechnung|derivative]] is obtained by elemental differentiation:

:&lt;math display=&quot;block&quot;&gt;
  
f^\prime(x) = \sum_{n=1}^\infty a_n n \left( x-x_0 \right)^{n-1}= \sum_{n=0}^\infty a_{n+1} \left(n+1 \right) \left( x-x_0 \right)^{n}

&lt;/math&gt;
&lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; can be differentiated as often as desired and the following applies:

:&lt;math display=&quot;block&quot;&gt;
  
f^{(k)}(x) = \sum_{n=k}^\infty \frac{n!}{(n-k)!} a_n (x-x_0)^{n-k}
           = \sum_{n=0}^\infty \frac{(n+k)!}{n!} a_{n+k} (x-x_0)^n

&lt;/math&gt;
Analogously, a [[w:de:Stammfunktion|antiderivative]] is obtained by means of a link-wise integration of a potency series:

:&lt;math display=&quot;block&quot;&gt;
  
\int f(x)\,\text{d}x = \sum_{n=0}^\infty \frac{a_n \left( x-x_0 \right)^{n+1}} {n+1} + C = \sum_{n=1}^\infty \frac{a_{n-1} \left( x-x_0 \right)^{n}} {n} + C

&lt;/math&gt;
In both cases, the convergence radius is equal to that of the original row.

==Presentation of functions as potency series==
Often, a given function is interested in a potency series representation – in particular to answer the question whether the function [[w:de:Analytische Funktion|analytic]] is. There are some strategies to determine a potential series representation, the most common by the [[w:de:Taylorreihe|Taylor series]]. Here, however, the problem often arises that one needs a closed representation for the discharges, which is often difficult to determine. However, there are some lighter strategies for [[w:de:gebrochen rationale Funktion|fuctional rational functions]]. As an example the function

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}
&lt;/math&gt;
to be considered.

===By means of the geometric series===
By factoring the denominator and subsequent use of the formula for the sum of a [[w:de:Geometrische Reihe|geometrischen Reihe]], a representation of the function as a product of infinite rows is obtained:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{(1-z)(3-z)}=\frac{z^2}{3}\cdot \frac{1}{1-z} \cdot \frac{1}{1-\frac{z}{3}} =  
&lt;/math&gt;

:&lt;math display=&quot;block&quot;&gt;
    = \frac{z^2}{3}\cdot \left(\sum_{n=0}^\infty z^n \right) \cdot \left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right)= \frac{1}{3}\left(\sum_{n=2}^\infty z^n \right)\left( \sum_{n=0}^\infty \left(\frac{z}{3}\right)^n\right) 
&lt;/math&gt;

===Product of geometric rows===
Both rows are potency rows around the development point &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; and can therefore be multiplied in the above-mentioned manner. The same result also provides the [[w:de:Cauchy-Produktformel|Cauchy-Product formula]]

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\sum_{n=0}^\infty \left(  \underbrace{\sum_{k=0}^n a_k b_{n-k}}_{c_n} \right) \cdot z^n 
&lt;/math&gt;
Series (mathematics)
===Coefficients of individual series===
The following shall apply:

:&lt;math display=&quot;block&quot;&gt;
  a_k=
\begin{cases}
0 &amp; \text{ für } k\in\{0,1\} \\
1 &amp; \text{ sonst}
\end{cases}
&lt;/math&gt;
and

:&lt;math display=&quot;block&quot;&gt;
  b_k=\frac{1}{3^k}.
&lt;/math&gt;

===Cauchy product formula===
This follows by applying the formula for the partial sum of a [[w:de:Geometrische Reihe|geometric series]]

:&lt;math display=&quot;block&quot;&gt;
  c_n = \sum_{k=0}^n a_k b_{n-k}= \sum_{k=2}^n \left( \frac{1}{3}\right)^{n-k}=\frac{1}{3^{n-2}}\sum_{k=0}^{n-2} 3^k= -\frac{1-3^{n-1}}{2 \cdot 3^{n-2}}
&lt;/math&gt;
as a closed representation for the coefficient sequence of the potency series. Thus, the potency series representation of the function around the development point 0 is given by

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \sum_{n=2}^\infty \frac{3}{2} \cdot \left(1-\frac{1}{3^{n-1}} \right) \cdot z^n
&lt;/math&gt;.

===Application of geometric rows or coefficient comparison===
As an alternative to geometrical series, it is an alternative to [[w:de:Koeffizientenvergleich|coefficient comparison]] is an alternative: One assumes that a power series representation exists for &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)= \frac{z^2}{z^2-4z+3}= \sum_{n=0}^\infty c_n z^n
&lt;/math&gt;
The function &lt;math display=&quot;inline&quot;&gt;
  f
&lt;/math&gt; has the unknown coefficient sequence &lt;math display=&quot;inline&quot;&gt;
  (c_n)_{n \in \mathbb{N}}
&lt;/math&gt;. After multiplication of the denominator and an index shift, the identity results:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} z^2 &amp; = (z^2-4z+3)\sum_{n=0}^\infty c_n z^n \\&amp; = \sum_{n=2}^\infty c_{n-2} z^n - \sum_{n=1}^\infty 4c_{n-1} z^n + \sum_{n=0}^\infty 3c_n z^n \\&amp; = 3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n \end{align}
&lt;/math&gt;
The potency series &lt;math display=&quot;inline&quot;&gt;
  g(z):=z^2Series (mathematics)
&lt;/math&gt; is compared with the potency series &lt;math display=&quot;inline&quot;&gt;
  h(z):=3c_0+z(3c_1-4c_0) + \sum_{n=2}^\infty (c_{n-2} -4 c_{n-1} +3 c_n)z^n
&lt;/math&gt;. Both potency rows have the same development point &lt;math display=&quot;inline&quot;&gt;
  z_o=0
&lt;/math&gt;. Therefore, the coefficients of both potency rows must also correspond. Thus, the coefficient of (698-1047-1731592552598-341-99 must be &lt;math display=&quot;inline&quot;&gt;
  0=3c_0
&lt;/math&gt;, for which the coefficient of &lt;math display=&quot;inline&quot;&gt;
  z^1
&lt;/math&gt; applies &lt;math display=&quot;inline&quot;&gt;
  0=3c_1-4c_0
&lt;/math&gt;, ...

===Recursion formula for coefficients===
However, since two potency rows are exactly the same when their coefficient sequences correspond, the coefficient comparison results

:&lt;math display=&quot;block&quot;&gt;
  c_0=0,\ c_1=0,\ c_2=\frac{1}{3}
&lt;/math&gt;
and the recursion equation

:&lt;math display=&quot;block&quot;&gt;
  r = \lim_{n\to\infty} \left| \frac{a_{n}}{a_{n+1}} \right|,
&lt;/math&gt;;
the above closed representation follows from the complete induction.

===Benefits coefficient comparison===
The method by means of coefficient comparison also has the advantage that other development points than &lt;math display=&quot;inline&quot;&gt;
  z_0=0
&lt;/math&gt; are possible. Consider the development point &lt;math display=&quot;inline&quot;&gt;
  z_1=-1 
&lt;/math&gt; as an example. First, the broken rational function must be shown as a polynomial in &lt;math display=&quot;inline&quot;&gt;
  (z-z_1)=(z+1)
&lt;/math&gt;:

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=\frac{(z+1)^2-2(z+1)+1}{(z+1)^2-6(z+1)+8}
&lt;/math&gt;

===Other points of development===
Analogously to the top, it is now assumed that a formal potency series around the development point exists with unknown coefficient sequence and multiplied by the denominator:

:&lt;math display=&quot;block&quot;&gt;
  \begin{align} (z+1)^2-2(z+1)+1 &amp; = ((z+1)^2 - 6(z+1)+8)\sum_{n=0}^\infty c_n (z+1)^n \\ &amp; = 8c_0+(z+1)(8c_1-6c_0) + \\ 
   &amp;  \qquad +  \sum_{n=2}^\infty (c_{n-2}-6c_{n-1}+8c_n)(z+1)^n \end{align}
&lt;/math&gt;
Again, by means of coefficient comparison

:&lt;math display=&quot;block&quot;&gt;
  c_0=\frac{1}{8},\ c_1=-\frac{5}{32},\ c_2=-\frac{1}{128}
&lt;/math&gt;
and as a recursion equation for the coefficients:

:&lt;math display=&quot;block&quot;&gt;
  c_n=\frac{-c_{n-2}+6c_{n-1}}{8}
&lt;/math&gt;

===Partial fraction decomposition===
If the given function is first applied [[w:de:Polynomdivision|Polynomial division]] and then [[w:de:Partialbruchzerlegung|Partial fraction decomposition]], the representation is obtained

:&lt;math display=&quot;block&quot;&gt;
  f(z)=\frac{z^2}{z^2-4z+3}=1+\frac{4z-3}{(z-1)(z-3)}= 1 + \frac{1}{2} \cdot \frac{1}{1-z} - \frac{3}{2} \cdot \frac{1}{1-\frac{z}{3}}
&lt;/math&gt;.
By inserting the geometric row, the following results:

:&lt;math display=&quot;block&quot;&gt;
  {f(z)=1+\frac{1}{2} \cdot \sum_{n=0}^\infty z^n - \frac{3}{2} \cdot \sum_{n=0}^\infty \frac{1}{3^n}z^n = 1+\sum_{n=0}^\infty \frac{3}{2} \cdot\left( 1- \frac{1}{3^{n-1}} \right) z^n}
&lt;/math&gt;
The first three sequence elements of the coefficient sequence are all zero, and the representation given here agrees with the upper one.

==Generalizations==
Potency rows can be defined not only for &lt;math display=&quot;inline&quot;&gt;
  x \in \mathbb{R}
&lt;/math&gt;, but are also generalizable. Thus, for example, R B is the [[w:de:Matrixexponential|Matrix exponential]] and the [[w:de:Matrixlogarithmus|Matrix logarithm]] generalizations of potency rows in the area of the [[w:de:Matrix (Mathematik)|square of matrices]].
If in a row also potencies with negative integer exponents occur, one speaks of a [[w:de:Laurent-Reihe|Laurent-series]]. If the exponent is allowed to accept broken values Series (mathematics), it is a [[w:de:Puiseux-Reihe|Puiseux-series]].
[[w:de:Formale Potenzreihe|Formal power series]] are used, for example, as [[w:de:erzeugende Funktion|generating Function]]s in [[w:de:Kombinatorik|combinations]] and [[w:de:Wahrscheinlichkeitstheorie|probabilitytheoery]] (for example [[w:de:wahrscheinlichkeitserzeugende Funktion|probability generating functions]]). In the [[w:de:Algebra|Algebra]], formal power series are examined over general [[w:de:Kommutativer Ring|complex ring]].

== References ==
* Kurt Endl, Wolfgang Luh: ''Analysis II.'' Aula-Verlag 1973, 7th edition 1989, ISBN 3-89104-455-0, pp. 85–89, 99.
* E. D. Solomentsev: [http://eom.springer.de/P/p074240.htm ''Power series.''] In: ''[[w:en:Encyclopedia of Mathematics|Encyclopedia of Mathematics]]''
&lt;references/&gt;

== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Power%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Power%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

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=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Potenzreihe Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* [[w:de:Kurs:Funktionentheorie/Potenzreihe|Kurs:Funktionentheorie/Potenzreihe]] URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Potenzreihe
* Date: 11/14/2024
* [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Potenzreihe&quot; srclang=&quot;de&quot; date=&quot;11/14/2024&quot; time=&quot;14:57&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Potenzreihe]]
&lt;/noinclude&gt;
&lt;!-- &lt;noinclude&gt;[[en:Course:Function theory/Potence series]]&lt;/noinclude&gt; --&gt;
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[[Category:Analytic functions|analytic function]]
[[Category:Sequences and series|Sequences]]
[[Category:Wiki2Reveal]]</text>
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  <page>
    <title>Happy haskell hours</title>
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      <text bytes="3528" sha1="dwmqp79o0ihqqcsqxv9n2ssxsc31e21" xml:space="preserve">=Happy Haskell Hours!=

This page documents my daily practice and exploration of Haskell and functional programming.

The purpose is consistency, clarity of thought, and long-term growth toward scientific computing in Haskell.

== Daily Log ==
A daily log of my ongoing practice and exploration of Haskell and functional programming. The purpose of maintaining this log is consistency rather than performance — to document what I study, what I implement, and what I understand each day. Over time, this record should reflect gradual progress toward building a strong foundation for scientific computing in Haskell.

[[Happy haskell hours/Daily log Feb March|Feb - March 2026]]

== Intend to Explore ==
# Course: Functional Programming by Graham Hutton
# Introduction to Haskell Course by &quot;Well Typed&quot;.
# Haskell For Dilettantes
# Books by Richard Bird
# Domain Specific Language for Mathematics Course.
# Competitive Programming
# Learn Physics with Functional Programming book.
# Scientific computing projects
# Panel methods
# FEM code
# Optimization algorithms
# Quantum computing
# linear and nonlinear solvers
# Numerical Methods implementation
# Numerical Analysis 
# Algorithms for FP
# Science of FP
# Functional data structures
# Courses by Patrik Jansson
# Finite difference solvers of CFD 
# Finite volume method
# Grid generation and adaptation algorithms in haskell
# https://petsc.org/release/tutorials/

== Researchers and Practitioners I learn from ==

# [https://simon.peytonjones.org/ Simon Peyton Jones.]
# [https://chalmersfp.github.io/members/rjmh.html John Hughes.]
# [https://www.cse.chalmers.se/~koen/ Koen Classen.]
# [https://simonmar.github.io/ Simon Marlow.]
# [https://patrikja.owlstown.net/ Patrick Jansson.]
# [https://martinescardo.github.io/ Martin Escardo.]
# [https://www.gla.ac.uk/schools/computing/staff/philtrinder/ Phil Thrinder.]
# [https://chalmersfp.github.io/members/ms.html Mary Sheeran.]
# [https://chalmersfp.github.io/members/secarl.html Carl-Johan Seger.]
# [https://bartoszmilewski.com/ Bartosz Milewski.]
# [https://www.uu.nl/staff/GKKeller Gabriele Kellar.]
# [https://justtesting.org/ Manual M T Chakravarty.]
# [https://www.ost.ch/en/person/farhad-d-mehta-8699 Farhad Mehta]
# [http://ozark.hendrix.edu/~yorgey/forest/index/index.xml Brent Yogrey]
# [https://www.andres-loeh.de/ Andres Loh]
# [https://sergey-goncharov.org/ Sergey Goncharov]
# [https://www.lvc.edu/profiles/dr-scott-n-walck/ Scott N Walck]
# [https://omelkonian.github.io/ Orestis Melkonian]
# [https://jmchapman.io/index.html James Chapman]
# [https://dylanbeattie.net/ Dylan beattie]
# [https://www.cs.ox.ac.uk/people/jeremy.gibbons/ Jeremy Gibbsons]
# [https://people.epfl.ch/daniel.kressner?lang=en Daniel Kressner] - Professor - EPFL - Numerical Algorithms and High Performance Computing
# [https://people.epfl.ch/laura.grigori?lang=en Laura Grigori] - Professor - EPFL - High Performance Numerical Algorithm
# [https://www.nicolasboumal.net/ Nicolas Boumal] - Professor - EPFL - Optimization
# [https://cm.dmi.unibas.ch/teaching/teaching.html]

== Websites &amp; Podcasts ==

# [https://chalmersfp.github.io/ Chalmers Functional Programming Group.]
# [https://haskell.foundation/podcast/ The Haskell Interlude.]
# [https://youtube.com/playlist?list=PLD8gywOEY4HaG5VSrKVnHxCptlJv2GAn7&amp;si=DKXSKaajeL7c9kNr Haskell Unfolder Podcast]
# Serokell youtube

== Hackage Libraries ==

# physics
# vector
# linear
# learn-physics
# dimensional
# chiphunk
# LPFP
# science-constants
# lin-alg
#

[[Category:Haskell]]</text>
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  <page>
    <title>Just sustainability transitions: a living review</title>
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      <contributor>
        <username>Amélie E. Pereira</username>
        <id>3042711</id>
      </contributor>
      <comment>/* Main subjects */</comment>
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      <text bytes="39090" sha1="nmk7iobnmw6td4wbpzebf97fcghmzaw" xml:space="preserve">== Introduction ==

=== Definition of living review ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot;/&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. 

[[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but &quot;not yet ready for use&quot;&lt;ref&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;.

===  Definitions of just transition : ===
* «a fair and equitable process of moving towards a post-carbon society’. »&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;.


The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 
 
=== Definition of Procedural justice ===
Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt;. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For example, the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;.     

== Methodology ==

=== Wikidata and the semantic web ===&lt;!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner
 --&gt;

=== Database search ===
We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article screening ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper

=== Importing selected articles into Wikidata ===
To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through meta-data enrichement ===&lt;!-- Add : What is meta-data enrichement --&gt;

Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).&lt;ref name=&quot;:5&quot; /&gt; We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus.

==== Main subjects ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
|-
|Q42377797
|
|-
|Q2798912
|
|-
|Q421953
|
|}&lt;!-- include all below items using the wikidata link template
 --&gt;
&lt;nowiki&gt;https://www.wikidata.org/wiki/Q84459973&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q185836&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4764988&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4338318&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4930066&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q430460&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q7569&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4116870&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q125928&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q260607&lt;/nowiki&gt;

&lt;nowiki&gt;https://en.wikipedia.org/wiki/Climate_change_mitigation&lt;/nowiki&gt;

Q1291678

Q2270945

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q16972712&lt;/nowiki&gt;

Q16324410

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q11024&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q177634&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q5154673&lt;/nowiki&gt;

Q113514984

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q65807646&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q188843&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q11693783&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q284289&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q7174&lt;/nowiki&gt;

Q552284

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1230584&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1049066&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q8134&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q295865&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1358789&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q868575&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q138359220&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q131444737&lt;/nowiki&gt;

www.wikidata.org/wiki/Q16869822

Q14944319

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q192704&lt;/nowiki&gt;

Q117091181

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q24965464&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1805337&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1341244&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q3406659&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q3456219&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q2700433&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q837718&lt;/nowiki&gt;

Q795757

Q795757

Q1479527

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q771773&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q56395513&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q5465532&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4421&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q48277&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1553864&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q8458?wprov=srpw1_0&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q11376059&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q103817&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q113561794&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q770480&lt;/nowiki&gt;

Q17142211

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1516555&lt;/nowiki&gt;

Q6316391

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q366139&lt;/nowiki&gt;

Q3027857

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q59679511&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q43619&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q127514833&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q13023682&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q728646&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q3907287&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q9357091&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q265425&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q25107&lt;/nowiki&gt;

Q442100

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q7249406&lt;/nowiki&gt;

Q7257735

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q541936&lt;/nowiki&gt;

Q6142016

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q10509953&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q12705&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q56510941&lt;/nowiki&gt;

Q1165392

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q4414036&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q17152351&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q187588&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q264892&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q34749&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q2930198&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q125359881&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q219416&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q131201&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q7649586&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q69883&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q920600&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q3376054&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q107389921&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q7981051&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q467&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q188867&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q1038171&lt;/nowiki&gt;

Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords.

==== Study types ====
Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&amp;property=P279&amp;mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : &lt;!-- include all below items using the wikidata link template
 --&gt; 

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q603441&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q472342&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q815382&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q1504425&lt;/nowiki&gt;

&lt;nowiki&gt;https://www.wikidata.org/wiki/Q2412849&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q6822263&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q7301211&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q17007303&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q70470634&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q101116078&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q110665014&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q137174203&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q137174450&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q137209848&lt;/nowiki&gt;

&lt;nowiki&gt;http://www.wikidata.org/entity/Q137211242&lt;/nowiki&gt;

[Include list and description of types of litterature reviews]

Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Research site ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

=== Knowledge modelling ===
Concept maps can also be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;.

We used the Author Disambiguator tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers: following a query, it categorises scientific publications into thematic groups. It also automatically searches for ORCID, ResearchGate and VIAF pages&lt;ref&gt;https://author-disambiguator.toolforge.org/&lt;/ref&gt;. 

==== Thematic networks ====
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as grounded theory&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes.  

[Illustration of thematic networks] 

Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified.

==== Conceptual modelling ====
Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt; we summarize these approaches below and examine which wikidata properties exist to represent them. 

* Definition: the content of a concept is formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
* Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}.
* Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}.
* Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be.
* Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

===== Difficulties encountered in modelling concepts: =====

*{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent.
* {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals

During the construction of the graph, the main difficulty encountered was the numerous bugs in the open-source tools used. For example, the &quot;Author Disambiguator&quot; tool, used to create entries for the researchers who worked on the publications we are analysing, fails to launch about half the time, displaying the message &quot;too many requests&quot;

The Orcidator tool, which, as its name suggests, automatically adds ORCIDs to researchers’ profiles, could not be used. The message &quot;DEACTIVATED BECAUSE OF ABUSE&quot; appears after logging&lt;ref&gt;https://sourcemd.toolforge.org/orcidator_old.php, tested on 7 April 2026&lt;/ref&gt;. 

==== Causal networks ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers often present models with boxes and arrows representing their hypothesis about how variables are expected to correlate{{Citation needed}}. Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;.

Wikidata includes several properties to describe causal relationships:
* {{Wikidata entity link|P828}}
* {{Wikidata entity link|P1542}}
* {{Wikidata entity link|P1537}}
* {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors
'''Chronologies'''

=== Writing ===
To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below.

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
| [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa
|-
| [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities
|-
| [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion
|-
| [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review
|-
| [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights
|-
| [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review
|-
| [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations
|-
| [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance
|-
| [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions
|-
| [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review
|-
| [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice
|-
| [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice
|-
| [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review
|-
| [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review
|-
| [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions
|-
| [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition
|-
| [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy
|-
| [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends
|-
| [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience
|}


== References ==
{{References}}</text>
      <sha1>nmk7iobnmw6td4wbpzebf97fcghmzaw</sha1>
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  </page>
  <page>
    <title>User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells/sandbox</title>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The perfection of Fuller's cyclic design */</comment>
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      <text bytes="220866" sha1="63fk4sllbew2o7fmpo6ptf9m81ijdsg" xml:space="preserve">== Bitruncating the 5-cells ==

{{Efn|There are 30 and only 30 distinct chordal distances between vertices in the 120-cell. The Petrie polygon of the 120-cell is a skew {30} triacontagon. The 120-cell's edge is the {30/1} chord. The 5-cell edge chord is the {30/8} chord, which connects two vertices that are 8 120-cell edges apart on the 30-edge Petrie polygon. In the regular 5-cell the characteristic isoclinic rotation in the edge planes which takes the 5 vertices to each other has period 15, and takes the 30 vertices of a compound of 6 disjoint regular 5-cells (called the [[W:Bitruncated 5-cell|bitruncated 5-cell]]) to each other. A regular 5-cell's 10 edges lie in 10 distinct {12} central planes, and the 6 5-cells of the compound share the same set of 10 central planes. The invariant planes of rotation are the 10 {12} central planes containing the 104.5° 5-cell edges, and the 41.1° isocline chords of the rotation are bitruncated 5-cell edges. In each step of the rotation a vertex moves 41.1° over a bitruncated 5-cell edge to a vertex position of a disjoint 5-cell in the compound. In the complementary characteristic isoclinic rotation of the bitruncated 5-cell, the invariant planes of rotation are 15 central planes containing the 41.1° bitruncated 5-cell edges (these are not {12} central planes), and the 75.5° isocline chords of the rotation lie in the 10 {12} central planes containing all the 104.5° 5-cell edges; they are the 180° complements of the 5-cell edge chords, and form great rectangles with them.|name=characteristic 5-cell rotation}}

It is difficult to illustrate the 30-vertex bitruncated 5-cell informatively by projecting it to its 3-dimensional polyhedral shadow. One projection of this hull is a regular icosahedron, but that is deceptive because there are actually 60 edges, not 30, and the 20 triangle faces only meet at their vertices, not at their edges. The triangle faces meet the hexagon faces at their edges. Another projection is an icosidodecahedron, which has 30 vertices but is also misleading because its pentagon faces are not faces of the 4-polytope and its cells at all. The hexagons which ''are'' faces are not visible in either of these projections.

== Truncations of the 5-cells ==
...

* Rectifying the radius 2{{radic|2}} 5-cell of edge 2{{radic|5}} creates the radius {{radic|2}} rectified 5-cell of edge {{radic|5}}.

* Rectifying the radius 2 5-cell of edge 2{{radic|5/2}} creates the unit-radius rectified 5-cell of edge {{radic|5/2}}.

* Rectifying the radius {{radic|2}} 5-cell of edge {{radic|5}} creates the radius {{radic|1/2}} rectified 5-cell of edge {{radic|5}} / 2.

* Rectifying the unit-radius 5-cell of edge {{radic|5/2}} creates the radius 1/2 rectified 5-cell of edge {{radic|5/2}} / 2.


...

== Questions regarding Blind's 11-point augmented rectified 5-cell ==
What happens when we rectify the 5-cells in the 120-cell? Rectifying the 5-cell creates a 10-point (10-cell) semi-regular 4-polytope with 30 equilateral triangle faces, known as the [[W:Rectified 5-cell|rectified 5-cell]]. If we truncate the 600 vertices of a 120-cell this way, we create new vertices at the mid-edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. In a 120-cell of radius {{radic|2}}, the smaller 4-polytope thus created will have a diameter of {{radic|3}}. The 120-cell contains 120 5-cells and many pairings of 5-cells inscribed in opposing positions, with a pair of antipodal vertices, edges, faces, or cells. Here we are concerned with pairs of 5-cells with a pair of antipodal edges. The 5-cell mid-edge to 5-cell mid-edge diameter of the smaller 3-sphere is the {{radic|3}} chord of 75.5°, the 180° complement of the {{radic|5}} 5-cell edge chord of 104.5°. These two chords lie in the 200 irregular {12} dodecagon central planes, where they form three great rectangles.

How many vertices does the smaller 4-polytope have? That is, how many 5-cell edge intersections occur in the 120-cell? The answer depends on whether or not the 120-cell contains pairs of 5-cells in dual position, the edges of which intersect. The intersection of a dual-position 5-cell pair is known to be a 30-point (10-cell) [[W:Bitruncated 5-cell|bitruncated 5-cell]], with 20 triangle faces and 20 hexagon faces.

If 5-cells in dual position exist in the 120-cell, do the 5-cell edges intersect orthogonally? If so, by symmetry do ''four'' 5-cells lie in dual position in the 120-cell (six pairs of 5-cells in dual position), with four orthogonal 5-cell edges at each intersection point, where none of the four orthogonal lines is an axis intersecting the center of the 120-cell?

We can see from the irregular {12} central plane chord diagram that 5-cell edges in different planes will not intersect each other at their mid-points; rather, each 104.5° 5-cell edge has two such points of intersection, 60° apart. The distance between them on the 5-cell edge is a 24-cell edge of the smaller 120-cell. Is a 24-cell edge of the smaller 120-cell colinear with the center portion of a 5-cell edge of the larger 120-cell? Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges meet in cubic vertex figures. The 24-cell's edges also meet in a cubic vertex figure. Are the {{radic|5}} 5-cell edges of the larger 120-cell the {{radic|1}} 24-cell edges of the smaller 120-cell, extended at both ends?

If four 5-cells lie in dual position with their 10 edges intersecting orthogonally at one of their two points of intersection, must the same four 5-cells also intersect orthogonally at the other point of intersection in each edge? Are the four disjoint 5-cells in dual-position Clifford parallel and one 24-cell edge apart? If the 120 5-cells lie in orthogonal dual position in groups of four, the 1200 5-cell edges with two intersection points each are reduced to 600 distinct vertices, making the smaller 4-polytope a smaller 120-cell.

Is the smaller 4-polytope whose vertices are the 5-cell mid-edge intersections also a 600-point 120-cell, so that the 120-cell is concentric to a smaller-radius instance of itself?{{Efn|The self-dual 24-cell is concentric to a smaller instance of itself of half its radius, which constitutes its [[24-cell#Relationships among interior polytopes|common core]], shared with its inscribed 8-cells and 16-cells.}} In that case the 10-point (10-cell) rectified 5-cell is found in the 120-cell, and its edges are chords joining the 600 vertices of the 120-cell: are they 24-cell edge chords? Where exactly in the 120-cell do we find the tetrahedral and octahedral cells of the rectified 5-cell? In particular, are the rectified 5-cell octahedral cells also 24-cell octahedral cells? {{Efn|The rectified 5-cell is said to be the vertex figure of the [[W:5-demicube|5-demihypercube]], the 5-dimensional analog of the 4-demihypercube (16-cell) and the 3-demihypercube (tetrahedron).}}

What is the geometric relationship between these two 10-cell 4-polytopes, the 30-point bitruncated 5-cell and the 10-point rectified 5-cell? Are their triangle faces congruent? The cells of the [[W:Bitruncated 5-cell|bitruncated 5-cell]] are [[W:Truncated tetrahedron|truncated tetrahedra]], with 4 triangle faces and 4 hexagon faces. Where in the 120-cell do we find these truncated tetrahedra? Are their triangle faces 24-cell faces? What are their hexagon faces?

Does the 120-cell also contain pairs of 24-cells in dual position, which intersect at their mid-edges? The mid-edge intersection points of 24-cells in dual position would lie on a third 3-sphere, intermediate in diameter between the original 120-cell and the smaller 120-cell. If this intermediate-size concentric 4-polytope exists within the 120-cell, how many vertices does it have? Do the 24-cell edges (which are also tesseract edges) intersect at their mid-points, and are they orthogonal? If so, by symmetry do ''three'' 24-cells lie in dual position in the 120-cell (three pairs of 24-cells in dual position), with three orthogonal intersecting edges, where the fourth orthogonal line at each intersection point is a 24-cell mid-edge to mid-edge diameter axis intersecting the center of the 120-cell?

The geometers who categorized the [[W:Blind polytope|Blind polytopes]] in 1979 discovered a semi-regular 11-point convex 4-polytope with 42 equilateral triangle faces and 36 edges, an [[W:Rectified 5-cell#Other forms|augmented rectified 5-cell]]. It is constructed by adding one point to the 10-point (10-cell) rectified 5-cell, raising an octahedral pyramid on one (actually on each) of the 10-cell's 5 octahedral facets, creating another semi-regular convex 4-polytope with 13 tetrahedral and 4 octahedral facets: a 17-cell. If the 10-point rectified 5-cell is found in the 120-cell, is this 11-point (17-cell) 4-polytope that is made from it also found in the 120-cell? If so, where exactly, and in what incidence? What is its relation to the tetrahedral and octahedral cells of the rectified 5-cell, and to the truncated tetrahedral cells of the bitruncated 5-cell? What is its relation to the 120-cell's 5-cell edge {30/11} chord, and the isoclinic rotation of period 30 over that chord, which takes hemi-icosahedral rhombicosidodecahedron cells of the 11-cell to each other? Is the 11-point Blind 4-polytope the concrete realization of the abstract 11-point 11-cell?

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:04, 12 April 2025 (UTC)

== Questions regarding Blind's 11-point augmented rectified 5-cell ==
The rectified 5-cell is a Blind 4-polytope, a convex 4-polytope whose cells are all regular polyhedra. The geometers who categorized the [[W:Blind polytope|Blind polytopes]] in 1979 also discovered an 11-point Blind 4-polytope with 36 edges and 42 equilateral triangle faces, the [[W:Rectified 5-cell#Other forms|augmented rectified 5-cell]]. It is constructed by adding one point to the 10-point (10-cell) [[w:Rectified_5-cell|rectified 5-cell]], raising an octahedral pyramid on one of the 10-cell's 5 octahedral facets, creating a 4-polytope with 13 tetrahedral and 4 octahedral facets: a 17-cell. 

Augmenting the rectified 5-cell in this way, with an 11th vertex above one the 5 octahedral cells, adds 6 identical edges, 12 identical triangle faces, and 7 identical tetrahedral cells. The 11th vertex is located above the center of the octahedral cell, displaced radially out of the octahedron's hyperplane. The octahedral pyramid thus formed is exactly filled by 8 regular tetrahedral cells, which meet at its center. It is similar to the 600-cell vertex, an icosahedral pyramid exactly filled by 20 regular tetrahedral cells which meet at its center. 

There are five octahedral cells, so there are five ways to augment a rectified 5-cell in this way. If we augment all five octahedral cells, the resulting 15-point Blind polytope has 60 edges, 90 triangle faces, and 45 tetrahedral cells. 

The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. The tetrahedral-octahedral honeycomb has a cuboctahedron vertex figure, so in the rectified 5-cell, 12 edges radiate in 3-space from each vertex. The infinite tetrahedral-octahedral honeycomb's vertex figure is a radially equilateral cuboctahedron, because its cell radius equals its edge length. But the rectified 5-cell is not radially equilateral: its radius and edge length are unequal, and the 12 radii of its cuboctahedral vertex figure do not meet at 60° angles. ....

Although the 10-point rectified 5-cell is not found in the 120-cell, is this 11-point (17-cell) 4-polytope that is made from it found in the 120-cell? Where exactly is it found, and in what incidence? What is its relation to the tetrahedral and octahedral cells of the [[w:Rectified_5-cell|rectified 5-cell]], and to the truncated tetrahedral cells of the [[w:Truncated_5-cell|truncated 5-cell]] and [[w:Bitruncated_5-cell|bitruncated 5-cell]]? 

What is its relation to the 120-cell's 5-cell edge {30/11} chord, and the isoclinic rotation of period 30 over that chord, which takes hemi-icosahedral rhombicosidodecahedron cells of the 11-cell to each other?

Is the 11-point Blind 4-polytope a realization of the abstract 11-point 11-cell?

== 11 ==
[[W:11 (number)#Mathematics|The number eleven]] is:

: The third superprime number, occupying a prime position in the sequence of primes:

:: (3, 5, 11, 17, ...) in ordinal prime positions (2nd, 3rd, 5th, 7th, ...).

: The twin prime (2 apart) of 7:

:: &lt;math&gt;7 + 2 = 11&lt;/math&gt;

: Forms a sexy pair (6 apart) with 5 and 17 (a sexy threesome?):

:: &lt;math&gt;5 + 6 = 11&lt;/math&gt;

:: &lt;math&gt;11 + 6 = 17&lt;/math&gt;

: At the center of the series of known [[W:Brocard's problem#Brown numbers|Brown numbers]]:

:: (4,5), (5,11), (7,71)

: which are pairs satisfying the Brocard &lt;math&gt;n!+1 = m^2&lt;/math&gt; equalities:

:: &lt;math&gt;4! + 1 = 5^2 = 25&lt;/math&gt; {{color|red|&lt;-- 24-cells}}

:: &lt;math&gt;5! + 1 = 11^2 = 121&lt;/math&gt; {{color|red|&lt;-- 11-cells}}

:: &lt;math&gt;7! + 1 = 71^2 = 5041&lt;/math&gt;

: The rows of [[W:Pascal's triangle|Pascal's triangle]] are the powers of 11:

:: The rows of Pascal's triangle are the base-10 place-value digits of {{red|&lt;math&gt;\color{red}11^n&lt;/math&gt; in row &lt;math&gt;\color{red}n+1&lt;/math&gt;}}. (In any base number system, the place-value digit composition of each row is divisible by 11.)

:: The rows of Pascal's triangle are the configuration incidence counts of the {{red|&lt;math&gt;\color{red}n&lt;/math&gt;-simplex in row &lt;math&gt;\color{red}n+2&lt;/math&gt;}}. If the counts in each row are assigned {{red|&lt;math&gt;\color{red}+&lt;/math&gt; and &lt;math&gt;\color{red}-&lt;/math&gt; signs for even and odd dimensionality}} elements, the counts in each row sum to 0.

== 137 ==
[[W:137 (number)#Mathematics|The number 137]] is:

: The sum / difference of factorials:

:: &lt;math&gt;137 = 5! + 4! - 3! - 2! + 1!&lt;/math&gt; {{color|red|&lt;-- &lt;math&gt;\color{red}+&lt;/math&gt; and &lt;math&gt;\color{red}-&lt;/math&gt; signs (of dimensionalities?)}}

: A [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form &lt;math&gt;4n+1&lt;/math&gt;, where &lt;math&gt;n=34&lt;/math&gt;:

:: &lt;math&gt;137=4\times 34+1&lt;/math&gt;

: which is the sum of two squares:

:: &lt;math&gt;11^{2}+4^{2} = (121+16)&lt;/math&gt; {{color|red|&lt;-- rotations? }}

: A combination of three terms:

:: &lt;math&gt;4^{3}+3^{4}-2^{3} = (64+81-8)&lt;/math&gt;

: the cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link).

== The 137-point ==

The [[W:Electron|Great Celebrity]] arrives at the party, famous for guarding the mystery of her affiliations.{{Sfn|Mac Gregor 137|2017|loc=''137 and the equation E&lt;sub&gt;w&lt;/sub&gt; + E&lt;sub&gt;z&lt;/sub&gt; = E&lt;sub&gt;top&lt;/sub&gt;''}} She was invited, of course, and we hoped she would put in an appearance, but never dreamed she would arrive so soon. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:17, 1 May 2024 (UTC)

== Four dimensions provide four distinct kinds of stickiness ==

&lt;blockquote&gt;All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.&lt;/blockquote&gt;
Consider these two distinct types of glue. Perhaps the strong force (in physics), which acts over a short distance, is a rubber joint gluing several 1-dimensional polytopes together at a point, and each weaker force (including electromagnetism and gravity), which acts over a longer distance, glues ''n''-dimensional polytopes together (where ''n'' inversely characterizes the strength of the force).
[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 20:52, 17 December 2024 (UTC)

== Chords ==
{| class=&quot;wikitable&quot;
!colspan=4|Central planes of the 120-cell
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
|align=center|{12}&lt;sub&gt;1&lt;/sub&gt;
|align=center|{10}
|align=center|{4}
|-
!style=&quot;white-space: nowrap;&quot;|Edge chords
|align=center|15.5° + 44.5° = 60°
|align=center|36°
|align=center|90°

|- 
!style=&quot;white-space: nowrap;&quot;|Incidence
|align=center|200
|align=center|720
|align=center|2250
|-
!style=&quot;white-space: nowrap;&quot;|
|[[File:Great (12) chords of radius √2.png|250px]]
|[[File:Great (10) chords of radius √2.png|250px]]
|[[File:Regular_star_polygon_30-7.svg|250px]]
|}

== Regular non-convex 4-polytopes and the 5/2 chord ==

There are 16 regular 4-polytopes, and all 16 are inscribed in the 120-cell. In the 6 convex regular 4-polytopes the symbols &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt; are integers: 3, 4, or 5. In the 10 non-convex regular 4-polytopes at least one of those symbols is the rational number 5/2, which we see realized as the #8 chord. That distinctive chord is the edge chord of the regular 5-cell and the 11-cell. It also occurs as an interior chord in the 120-cell and in non-convex regular 4-polytopes, but it does not occur at all in the other 4 regular convex 4-polytopes: the 16-cell, 8-cell, 24-cell or 600-cell.

The #8 5/2 chord does not form a regular flat polygon in a central plane. It forms triangular face planes (in the 5-cell and 11-cell), but those triangles do not lie in central planes. The #8 chord forms ''skew'' {5/2} pentagrams in the 5-cell, with each edge of the pentagram lying in a distinct central plane. There are no ''flat'' {5/2} pentagrams of #8 chords anywhere in the regular 4-polytopes. Some of the non-convex regular 4-polytopes do have flat {5/2} pentagram faces, but their edges are not #8 chords: they are #9 chords of length &lt;math&gt;\phi&lt;/math&gt;. In the regular 4-polytopes, flat #8 pentagrams are illusory; they do not actually occur. Where we may think we see them, that is only a projection (flattening) to a 2-dimensional or 3-dimensional perspective view of a skew pentagram, which is actually a 4-dimensional object. It winds spread-out through all 4 dimensions, twice around the 3-sphere ''without self-intersecting'', before closing its circle.

== Alicia Boole Stott's original formulation of dimensional analogy ==
[[File:The Roots of Three.png|thumb|400px|Alicia Boole Stott's 1910 paper defined operations of expansion and contraction on regular polytopes, beginning with expansion of the 2-simplex equilateral triangle to a regular hexagon.]]
...

If we apply the Boole Stott expansion and contraction operations more generally, by relaxing the requirement that only a single edge length be involved, we can also describe the relationship between polytopes of different edge lengths as instances of expansion or contraction. For example, a unit-edge-length 24-cell can be reached as an expansion of a 16-cell of edge length {{radic|2}}, where the expansion distance between edges is also {{radic|2}}. In 3-sphere space this expansion can be performed in two chiral ways: left- or right-handed. Three concentric 16-cells result, inscribed in a {{radic|1}} edge-length 24-cell, if the expansion is done both ways. Corresponding pairs of their vertices are {{radic|3}} apart, corresponding pairs of their edges are {{radic|2}} apart, corresponding pairs of their cell centers are {{radic|1}} apart, and the 16-cell centers are {{radic|0}} apart.

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:12, 13 July 2024 (UTC)

== 5-cell ==
The 5-cell's &lt;math&gt;A_5&lt;/math&gt; symmetry group is simply the alternating group of 5 elements. &lt;math&gt;A_5&lt;/math&gt; is exceptional in that it is not only the symmetry group of the 4-simplex, it is also the symmetry group of the icosahedron and dodecahedron 3-polytopes.

&lt;blockquote&gt;The rotational symmetry group of the ''n''-simplex (not permitting reflections) is always the alternating group &lt;math&gt;A_n&lt;/math&gt;. When &lt;math&gt;n = 4&lt;/math&gt;, this coincides with the rotational symmetry group of the icosahedron and dodecahedron, which is also &lt;math&gt;A_5&lt;/math&gt;. (I believe this is the only &quot;exceptional isomorphism&quot; between symmetry groups of regular polytopes in any dimension, i.e. one besides the isomorphism between dual polytopes.)

Obviously one can prove this fact by showing that each group happens to be the alternating group on 5 elements, but I am curious whether there is a geometric approach to exhibiting the isomorphism - some natural bijection between rotations of the 4-simplex and rotations of the icosahedron which shows they exhibit the same multiplicative structure, without determining what that structure is. 

- Question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com], 2021&lt;/blockquote&gt;

There is some exceptional kind of inter-dimensional duality between the icosahedron and the 5-cell because of their common &lt;math&gt;A_5&lt;/math&gt; symmetry.

An icosahedron can be inscribed in a tetrahedron such that four icosahedron faces lie in the tetrahedron face planes. An author answering the above question on math.stackechange.com circumscribes five rotated regular tetrahedra around an icosahedron, with their 20 faces in the 20 face planes of the icosahedron, in order to demonstrate that the 5-cell and icosahedron belong to the same &lt;math&gt;A_5&lt;/math&gt; symmetry group. The five rotated tetrahedra are not a regular 5-cell, since they they are not face-bonded to each other, and not a 4-polytope since they all lie in the same hyperplane, but by considering their rotations the author concludes that the 5-cell and the icosahedron have the same &lt;math&gt;A_5&lt;/math&gt; symmetry.

A congruence of this kind between some 12-point (icosahedral) actual 4-polytope and a regular 5-cell must occur in 4-space. We can see where it must share at least one 5-cell face triangle with the 120-cell's rhombicosadodecahedral 8th section beginning with a cell. For each pairing of one of the 60 rhombicosadodecahedral central sections with one of the 20 disjoint 5-cells from which it contains a single inscribed face, it must be possible to identify a smaller, concentric 12-point (icosahedral) 4-polytope with one face in the 5-cell's face's plane. Do not this icosahedron's 20 faces lie in the 10 face planes of the same 5-cell, or in the 10 face planes of its completely orthogonal 5-cell? If we project this smaller icosahedron outward onto the 120-cell, it identifies 12 of the 120-cell vertices that form a regular icosahedron. They are also 12 of the rhombicosadodecahedron's 60 vertices, one from each of its 12 pentagon faces.

== Shlafli's criterion for the regular polytopes ==

== Heptad workspace architecture ==

[[File:Workspace architecture.png|thumb|Heptad round conference tables, hexad cubicals, tetrad corridors, and disjoint pentad digon structural pillars that pass through the tetrad voids between the hexad cubicals.]]
Since long before covid I have worked at home in my study, a square outbuilding that has one concave corner. I no longer go in to an office to work, but my ideal imaginary office cubicals are irregular hexagons abutting a square closet or structural pillar, and opening onto a shared work area with a round table. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 13:48, 4 May 2024 (UTC)

== Pentads and hexads together ==
In all three building cycles, in each step where 4-polytope building blocks are joined together to make a larger 4-polytope, an expansion of some 3-polytope embedded at the center of the 3-sphere is taking place, in which the 3-polytope's points are split into 2, 3, 4, or 5 points each and moved apart uniformly, creating a dimensionally analogous 4-polytope. Alicia Boole Stott invented these uniform expansion operations, and explored various distinct forms of them which turn points into edges, edges into faces, faces into cells, and do combinations of more than one of these things. Her goal was to discover new semi-regular 4-polytopes, by starting with the six regular 4-polytopes which were known and performing expansion and contraction operations on them. She began by studying the dimensional analogies between the five regular polyhedra and their regular 4-polytope analogues, and working out the 3D sections of the regular 4D polytopes, but found many more ways to perform expansions and contractions that paired a regular polytope with a semi-regular analogue. There was even a sixth regular 4-polytope which had no regular polyhedron it was an expansion of: the 24-cell. She found the 24-cell was an expansion of its central section, which is a semi-regular 3-polytope: the cuboctahedron. By working in both directions, she found all 15 Archimedean semi-regular 3-polytopes as contractions of regular 4-polytopes, and discovered their 45 Archimedean semi-regular 4-polytope analogues.

Heinz Hopf discovered the general form of these dimensional analogy expansion-contraction operations, which is that every point in the original 3-polytope expands into a linked great circle polygon in the resulting 4-polytope. Each uniform expansion operation determines a distinct dimensional analogy between some 3-polytope Hopf map and 4-polytope Hopf fiber bundle of linked great circle polygon fibers, but every uniform expansion operation turns every point in the 3-polytope into a polygon in a dimensionally analogous 4-polytope, where the polygons are all disjoint but linked together, each passing through all the others. That, it turns out, is the general nature of uniform expansion operations on the 3-sphere. Of course they can be run backwards, as contraction operations, and sometimes a construction is not possible as a straightforward expansion operation (there is no way to get there by expansion alone) but the dimensionally analogous result can be reached by expanding beyond it, and then contracting to it. There is a complexity that arises in doing so, however, because expansion and contraction operations can be paired in two chiral ways (left-handed and right-handed), since they do not commute. Expanding first and then contracting produces a different result than contracting first and then expanding, for the exact same pair of operations on the same object.

== Sequencing the 4-polytopes by complementary chord pairs ==

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=7|Sequencing the 4-polytopes by complementary chord pairs
|-
!
!
!pentads
!hexads
!heptads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!pentads
!hexads
!heptads
!
!
|}

See also the elements, properties and metrics of the sequence of convex 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:33, 23 May 2024 (UTC)

== Progression of the rational numbers as polytopes ==

The rational numbers are described as unit-radius regular 4-polytopes inscribed concentric to the 3-sphere. Except where explicitly described otherwise, polygons are regular great polygons in a central plane, not face polygons; regular polyhedra are inscribed at the center of the 3-sphere, not as cells or vertex figures; and 4-polytopes are one torus of the face-bonded cells of a regular convex 4-polytope, not the complete honeycomb that includes all its cells.

The 0-point {0} has no facets, and no extent. The 1-point {1} unigon has 1 non-facet a {{radic|1}} 0-point, and no 0-extent. The 2-point {2} digon has 2 0-facets each a {{radic|2}} unigon, and 1-extent {{radic|2&lt;sup&gt;2&lt;/sup&gt;/1&lt;sup&gt;4&lt;/sup&gt;}} = 2. The 3-point trigon {3} has 3 1-facets each a {{radic|3}} digon, and 2-extent {{radic|3&lt;sup&gt;3&lt;/sup&gt;/2&lt;sup&gt;4&lt;/sup&gt;}} = {{radic|27/16}} = 1.299. The 4-point quadragon {4} has 4 1-facets each a {{radic|2}} digon and 2-extent {{radic|4&lt;sup&gt;3&lt;/sup&gt;/2&lt;sup&gt;4&lt;/sup&gt;}} = {{radic|64/16}} = 2. It also has 2 diagonals each a unit-radius digon (an axis diameter) whose 1-extent is equal to that 2-extent. Thus it has 6 digons making an irregular flat tetrahedron with 4 non-disjoint right-triangle faces lying in one central plane.

The 4-point tetrahedron has 4 2-facets each a {{radic|5/2}} trigon and 3-extent {{radic|4/3}}; 6 1-facets each a {{radic|..}} digon. The 5-point 5-cell has .... [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:44, 28 May 2024 (UTC)

= A symmetrical arrangement of eleven 11-cells (draft notes by section) =

== 5-cells and hemi-icosahedra in the 11-cell ==

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} &lt;s&gt;The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell.&lt;/s&gt; The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron.

''prior to'' [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:42, 18 December 2024 (UTC)

== The eleventh chord ==

That projection from a {30/10} chord in the stationary 4-dimensional reference frame to a {30/8} chord in the rotating 2-dimensional reference frame, of a moving invariant plane as it tilts sideways in 4-space, is a [[W:Lorentz transformation|Lorentz transformation]]. It is the equation which relates an actual chordal distance within a 4-polytope to the foreshortened chord which it appears to be when seen in perspective view, as a 3-polytope or 2-polytope. The same relation expresses the equivalence (and equal validity) of two different systems of measurement, one in a rotating reference frame and one in its stationary reference frame, even though they yield distinctly different lengths.
[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:10, 7 March 2025 (UTC)

== Compounds in the 120-cell ==

...

=== Building the building blocks themselves ===

...

And here at last with the pentad and hexad orthoschemes we must be able to find formulae describing each polytope that relate &lt;math&gt;pentads^4 = hexads^4&lt;/math&gt;, the two equivalent constructions from root systems for every 4-polytope, even for polytopes which are obviously constructed one way, and not so obviously the other. One possible construction is always by pentad orthoschemes, since it is only necessary to construct the polytope's characteristic 4-orthoscheme, and 4-orthoschemes are pentads themselves. The other is by the hexads of the 16-cell, since 16-cells compound into everything larger. Surely every uniform 4-polytope can be constructed by some function of either of these root systems, however indirect the recipe. The expressions to do so then, with an equal sign between them, make a conservation law defining the 4-polytope, which we may call its physics by Noether's theorem. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 07:11, 26 April 2024 (UTC)

...

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

...see also the 120 right-triangle-faced, 62-point [[W:Disdyakis triacontahedron|disdyakis triacontahedron]]. As the barycentric subdivision of the regular dodecahedron and icosahedron, it contains the decomposition of all the &lt;math&gt;H_4&lt;/math&gt; polytopes to orthoschemes. It has a close relationship to the 60-point [[W:Rhombic triacontahedron|rhombic triacontahedron]]....

=== 120 of them ===
The{{radic|2}}-radius 120-cell has 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles. It contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== What's in the box ===
This is the picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell, including all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks &lt;s&gt;by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures&lt;/s&gt;.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). &lt;s&gt;The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.&lt;/s&gt;{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the torus decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers cited in the Wikipedia article make clear, but his description of its multi-faceted geometry is also the clearest set of instructions available for how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct manner in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a rotation of a distinct 4-polytope on the 3-sphere. The crucial point is that it is the rotation object in 4-space which is the real object, and the map in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they cannot justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a 120-cell cell ring (each fiber in the fiber bundle) is conflated to one distinct point on a 12-point (regular icosahedron) map. The characteristic rotation of the 120-cell which takes dodecahedral cells to each other within their respective rings of 10 dodecahedra is described by this Hopf map. Within the 120-cell (the territory mapped) we find an distinct instance of this map for every instance of the rotation it describes. The 120-cell contains 10 600-cells, each of which has 120 vertices, each with a regular icosahedron vertex figure that is the Hopf map of a distinct rotation.

In like fashion, we can look within the 120-cell for the Hopf maps of the 11-cell's characteristic rotations. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure, in complement to the way Moxness's 60-point rhombicosadodecahedron revealed their internal structure. We have already identified the 11-cell's characteristic rotation, in which the 11 vertices circulate on 11 disjoint {30/11} skew polygrams. The Hopf map of this rotation of the real 11-cell 4-polytope will be some 3-polytope that is an abstraction of it. Of course that abstract map must be the abstract 3-polytope we began with, the hemi-icosahedron, or more precisely, its realization as Moxness's 60-point rhombicosadodecahedron 3-polytope.

The obvious place to look for the 11-cell's Hopf map is near Moxness's 60-point rhombicosadodecahedron; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron and consider it as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that regular dodecahedra form 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. &lt;s&gt;But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere&lt;/s&gt;.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The rational numbers are each a distinct flavor.

Each natural number has its own distinct taste. The first six {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are prime integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is just mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. Each distinct #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord is some unique rational number &lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;. The #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f(h)+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons (not all shown here): a compound of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons), plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the #11 chord, shown with a non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram so that it lies parallel to the #8 chord. These two chords actually do lie parallel in the same {12} central planes.]]

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct rational numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity (as we saw [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|above]]), they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

These 15 chords (rational numbers) include the natural numbers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the ratios 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions, as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 9 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly, unless you have Steinbach's formula:
{| style=&quot;background-color:transparent; white-space:nowrap; float:left;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}
The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'' before closing their circuit, a helical geodesic circle or ''isocline'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a rational number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the rational number.

{| class=wikitable style=&quot;white-space:nowrap;&quot;
!colspan=6|Rational number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{3,\tfrac{5}{2}\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct rational number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the rational number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The real 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. &lt;s&gt;The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra.&lt;/s&gt; The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces &lt;s&gt;which lie between three cells&lt;/s&gt;, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

&lt;s&gt;Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|&lt;s&gt;Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves two ways: as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.&lt;/s&gt;|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.&lt;/s&gt;

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-hypercubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-hypercubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

== The 11-cell's real elements ==

We seek an object in the 120-cell which is the realization of the abstract 11-cell. The real 11-cell has central section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra, while the abstract 11-cell has hemi-icosahedron cells. How are the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra different than the abstract hemi-icosahedra?

Consider their incidences, previously described: 

* Each 11-cell contains 11 of the 120-cell's 120 completely disjoint 5-cells.
* Each hemi-icosahedron shares a face with 10 distinct 5-cells, and is completely disjoint from 1 of the 5-cells. Each 5-cell shares 1 of its 10 faces with each of 10 hemi-icosahedra, and is completely disjoint from 1 hemi-icosahedron.
* Each rhombicosidodecahedron shares a face with 20 distinct 5-cells, and is completely disjoint from 100 of the 5-cells.{{Efn|Each rhombicosidodecahedron contains only 20 5-cell faces, each from a distinct 5-cell, so it is face-disjoint from 100 of the 5-cells. It contains only 60 5-cell edges, so it is edge-disjoint from those 100 5-cells. It contains only 60 5-cell vertices, so it is vertex-disjoint from those 100 5-cells.}} Each 5-cell shares 1 of its 10 faces with each of 10 rhombicosidodecahedra, and is completely disjoint from 50 rhombicosidodecahedra.
* Each hemi-icosahedron shares its 10 faces (each containing 3 distinct 5-cell edges) with 10 other hemi-icosahedra.
* Each rhombicosidodecahedron shares its 10 {12} central planes (each containing 6 disjoint 5-cell edges) with 10 other rhombicosidodecahedra.

The abstract hemi-icosahedron's 5-cell faces do not actually meet at an edge; each hemi-icosahedron face is actually two 5-cell faces in different places; and 5-cell faces are disjoint in the rhombicosidodecahedra and in the 11-cell. Clearly, the 5-cell edges are not the 11-cell's real edges. The actual 11-cell edge chord is not the 5-cell edge {30/8} chord of length {{radic|5}}, but the longer {30/10) chord of length {{radic|6}}. This [[24-cell#Helical hexagrams and their isoclines|isocline chord of the 24-cell]] is the ''rotation'' e''dge chord'' of the 11-cell.

The 5-cell faces are not the real 11-cell's faces. The true face of the 11-cell is a great triangle of edge length {{radic|6}} (shown in green in the illustration), lying in a {12} central plane. The green great triangle is where two rhombicosidodecahedrons meet face-to-face as cells do. In the six regular convex 4-polytopes, 3-polytope cells meet pairwise at non-central face planes, and form a honeycomb of voluemtrically disjoint cells. Additionally, four of these regular 4-polytopes (all but the two simplest, the 5-cell and 16-cell) are compounds of smaller regular 4-polytopes, which meet pairwise at central planes. The simplest case is the 16-point (8-cell) tesseract, a compound of two 8-point (16-cells), which meet 2 at a {6} hexagon central plane. The 24-point (24-cell) is a non-disjoint compound of three 16-point (8-cell) tesseracts, which meet 2 at a {6} hexagon central plane. The 120-point (600-cell) is a compound of 5 disjoint 24-point 24-cells (ten different ways), which meet 2 at a {6} hexagon central plane. The 600-point (120-cell) is a compound of 5 disjoint 120-point 600-cells, two different ways, which meet 2 at a {12} dodecagon central plane. All these regular convex 4-polytopes are both a honeycomb of volumetrically disjoint 3-polytope cells, and they also form &quot;inside out honeycomb&quot; compounds of smaller 4-polytope &quot;hypercells&quot;, which have their shared &quot;faces&quot; on the inside, in their central rotation planes.

Like these other 4-polytopes, the 11-cell is both an assembly of volumetrically disjoint 3-polytope cells (they are the tetrahedral cells of its 11 5-cells), -- no it is not one assembly the 5-cells are disjoint -- and also a compound of non-disjoint 4-polytope &quot;hypercalls&quot; meeting 2 at a central plane. Each of its 4-polytope &quot;hypercells&quot; is -- what object exactly? -- comprised of selected parts of two disjoint polyhedra in different places: a completely orthogonal pair of 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedrons. Together those selected parts occupy all 4 dimensions, not just a 3-dimensional polyhedral section, and comprise an 11-point, 10-edge, 15-face 4-polytope. -- doesn't it have to be 15-edge, 10-face? -- The 11-cell is the non-disjoint compound of 11 of these 11-point 4-polytope &quot;hypercells&quot;.

The real 11-cell face is a {{Radic|6}} triangle in the {12} central plane parallel to two 5-cell face planes, which lie {{Radic|3}} above and {{Radic|3}} below it. The orthogonal distance between the two 5-cell faces is another {30/10} chord of length {{Radic|6}}, which joins their opposing vertices. That chord is another great triangle edge, in another {12} central plane, but it does not intersect any vertex of the parallel {12} central plane.

We found previously that the abstract hemi-icosahedron face was a {{radic|5}} 5-cell face; then that it was two 5-cell faces in disjoint 5-cells; then that it was really a {{radic|6}} great triangle in a central plane. These findings are not contradictory. The abstract hemi-icosahedron face is realized in a real rhombicosidodecahedron as a rigid object composed of these three parallel triangles. We will visualize that 9-point polyhedron precisely later, but hold the thought that an abstract 11-cell ''face'' is really a ''polyhedron''. This makes sense since the abstract 11-cell ''cell'' is really a ''4-polytope''. Once again, the 11-cell has something to teach us about interdimensional relationships, and the nature of dimensional analogy. -- where am I going with this suspect notion? --

The {12} central plane contains 4 disjoint great triangles (11-cell faces) in distinct rotational orientations, from 4 distinct 11-cells (although to reduce clutter only one of them is shown in green in the illustration). The {12} is the central plane of four distinct 9-point polyhedra, because it contains four distinct great triangles. The two parallel 5-cell face planes also contain triangles (5-cell faces), but these face planes are only {6}s, not {12}s. Each face plane contains only two 5-cell face triangles in opposing rotational orientations, not four triangles, because two of the four distinct 9-point polyhedra share a 5-cell face, the same way two tetrahedral cells of the 5-cell share the face. Pairs of distinct 9-point polyhedra form a 12-point polyhedron with two opposing 5-cell faces, parallel to a hexagon in the central plane between them.{{Efn|There are two regular great hexagons of edge length {{radic|2}} inscribed in each irregular {12} great dodecagon, although to reduce clutter they are not shown explicitly in this illustration. They are the famous [[24-cell#Great hexagons|24-cell great hexagons]]; each 24-cell has 16 of them. Each has two {{radic|6}} [[24-cell#Great triangles|great triangles]] inscribed in it.}} In each of 10 sets of three parallel planes (two 5-cell face planes on either side of a {12} central plane), we find four 9-point polyhedra (four sets of three parallel triangles), paired as two 12-point polyhedra (two sets of two parallel triangles with a parallel hexagon between them). The two opposing 5-cell faces in each {6} face plane belong to two disjoint 5-cells, and also to two distinct rhombicosidodecahedra. The 10 9-point polyhedra are inscribed in a single rhombicosidodecahedron, but the 12-point polyhedra are not.

...

== The 11-cell rotation ==

Eleven rhombicosidodecahedra are pairwise adjacent in two ways: (1) they are in contact, sharing central planes pairwise, similar to the way two cells of a 4-polytope are bound together at a shared face, and (2) they occupy adjacent Clifford parallel rotational positions in various 30-position isoclinic rotations, not in contact at all, but separated at every vertex by the same small distance. For each pair of rhombicosidodecahedra, these two kinds of adjacency occur in different places. There are places where they share a {{Radic|6}} triangle in a central plane, and other places where their two {{Radic|6}} triangles lie nearby, Clifford parallel, in adjacent positions of an isoclinic rotation which takes them to each other. The rotation takes entire rhombicosidodecahedra to each other's positions, in fact it takes all 60 rhombicosidodecahedra at once to each other's positions; an isoclinic rotation moves everything at once, altogether in parallel, in many different orthogonal directions at once. 

This is rather confusing to visualize, but let us try. In this case, the completely orthogonal invariant planes of the isoclinic rotation are pairs of {12} central planes. The 120-cell's 100 completely orthogonal pairs of {12} central planes all rotate at once, by the same displacement angle in each step of the rotation. In this case the angle is 120°, the arc of one edge of a {{Radic|6}} triangle in each central plane. Visualize 800 spherical triangles (or 200 irregular dodecagons, since there are 4 great triangles in each {12} central plane), all rotating like wheels. Now consider that they are 100 completely orthogonal pairs of wheels. Each is not only rotating like a wheel, it is also being rotated sideways like a coin flipping, each vertex moving on a sideways-moving rotating wheel. The actual path through space of each vertex is a closed spiral, a circular helix.

We are visualizing this as a coin rotating like a wheel while it flips sideways in space at the same time, because that is the closest thing to it we have observed in life, but it is more than that. A coin's twisting rotation takes place only in three dimensions: as a wheel in one 2-dimensional plane, and as a flipped coin in another 2-dimensional plane, but those two planes share an axis, because in 3-space there are only 3 orthogonal axes and 3 orthogonal planes through a central point. In 4-space there are 4 orthogonal axes and 6 orthogonal planes through a point (including 3 pairs of [[W:Completely orthogonal|completely orthogonal]] planes which intersect only at the central point, not anywhere in a line). 4-space is potentially very confusing, but the essential thing to understand about its 4 axes is that 4-space is simply roomier. It is more commodious than the 3-space we are confined in, and it has room to spread out in surprising new ways, such as the way two orthogonal circles need not be crammed into sharing an axis, as they must in 3-space. What this means for our visualization process is that we must relax our cramped view of space, and try to see the two completely orthogonal sideways-moving wheels correctly: each occupies a fully independent 2-axis 2-dimensional plane. It is easier to picture their motions ''incorrectly'', the way they would have to happen in 3-space, but a double rotation in 4-space, which we are not in the habit of visualizing correctly because it is unprecedented in our three-dimensional experience, is nonetheless accessible to the human visual imagination, which is a very powerful image processing engine that understands dimensional analogy really well.

...

== The 11-cell Hopf fibration ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the torus decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers cited in the Wikipedia article make clear, but his description of its multi-faceted geometry is also the clearest set of instructions available for how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct manner in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a rotation of a discrete 4-polytope on the 3-sphere. The crucial point is that it is the rotation object in 4-space which is the real object, and the map in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they cannot justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a 120-cell ring of dodecahedra (each fiber in the fiber bundle) is conflated to one distinct point on a 12-point (regular icosahedron) map. The characteristic rotation of the 120-cell which takes dodecahedral cells to each other within their respective rings of 10 dodecahedra is described by this Hopf map. Within the 120-cell (the territory mapped) we find an distinct instance of this map for every instance of the rotation it describes. The 120-cell contains 10 600-cells, each of which has 120 vertices, each with a regular icosahedron vertex figure that is the Hopf map of a distinct isoclinic rotation.

In like fashion, we can look within the 120-cell for the Hopf maps of the 11-cell's characteristic rotations. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure, in complement to the way Moxness's Hull #8 rhombicosidodecahedron reveals the cell's internal structure. We have already identified the 11-cell's characteristic rotation, in which the 11 vertices circulate on 11 disjoint {30/11} skew polygrams with {30/11} chord edges. The Hopf map of this rotation of the real 11-cell 4-polytope will be some 3-polytope that is an abstraction of it. Of course that abstract map must be the abstract 3-polytope we began with, the hemi-icosahedron, or more precisely, its realization as Moxness's Hull #8 rhombicosidodecahedron 3-polytope.

The rhombicosidodecahedron map has 60 vertices, and each vertex must ''lift'' to a disjoint great circle polygon of the 11-cell. The great circle polygons of the 11-cell rotation are its rotation edge polygons, which we found are {{Radic|6}} great triangles with {30/10} chord edges. These 60 disjoint great circle polygons must exactly fill the real 11-cell 4-polytope, comprising all its vertices. Thus the real 11-cell has 180 vertices. The 11 abstract vertices circulate on 11 disjoint 30-position {30/11} skew polygram isoclines, each abstract vertex visiting one-sixth {30} of the 180 vertices during the rotation. Each abstract vertex is a conflation of .. real vertices on .. {30/11} isoclines, and is represented at one time or another during the rotation by .. distinct vertices which its rotating real vertices visit. There are 11 discrete Hopf fibrations of the 11-cell, with 60 disjoint great {3} triangle fibers each, and 11 disjoint skew {30/11} triacontagram isocline fibers each. Each fibration corresponds to a distinct left (and right) instance of the characteristic isoclinic rotation.

...

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. &lt;s&gt;The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since&lt;/s&gt; &lt;s&gt;the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy&lt;/s&gt;

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cube). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}. Here we give two illustrations of the Jessen's, using different metrics: the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius. The correspondence between our 3-space and 4-space metrics is curiously cyclic; in the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra, in three different rhombicosidodecahedra, and like most polyhedron faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell ''faces'' at all, but central polygons of the cell. Of course they are the {{radic|6}} triangles seen in the irregular {12} dodecagon central planes, which we saw are the actual shared &quot;faces&quot; between adjacent hemi-icosahedron cells. Opposing {{radic|6}} triangles lie in completely orthogonal {12} central planes, where they are inscribed in great hexagons (but not in the same great hexagon of course, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] {12} planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The {{radic|5}} 5-cell faces do not appear in this illustration, only some of their edges do. The Jessen's (the building block we found 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 isoclinic cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being isoclinic rotations of each other. 

...

...removed the following from a footnote to the article, where it refers to the 5-cell edges that meet at 120-cell pentagon faces, because I am not sure it is true; possibly only the 11-cell edges, which meet at triangle faces, can act as the struts of a stable tensegrity structure (since pentagons unlike triangles are not rigid structures)
&lt;blockquote&gt;Consequently the 120-cell can be constructed as an infinitesimally mobile rigid geodesic 3-sphere: a 4-dimensional tensegrity sphere. The 120-cell's 1200 edges need only be tension cables, provided that a disjoint 600 of the 120 5-cells' 1200 edges are included as compression struts, in parallel pairs.|name=tensegrity 120-cell
&lt;/blockquote&gt;

...

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* Wright's architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', &lt;s&gt;an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}&lt;/s&gt;
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-hypercube), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-hypercube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-hypercubes), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==

The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its 600 instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton honeycomb 4-polytope or a non-disjoint compound 4-polytope, unless we call it both. We should not even firmly ascribe to it a distinct 4-dimensionality, since its 137-point compound contains or induces instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but a central point about them is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} Stott applied her operations to uniform polytopes of only one edge length recursively, starting with the regular polytopes, which did not lead her to the 11-cell, even though it has only one edge length. It is only because she did not have the 11-cell to start with that she did not discover the 137-point (..-cell) by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes, and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries, the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all these things singly, like the regular convex 4-polytopes, it compounds to significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has its realization in Euclidean 4-space as this convex 4-polytope with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, just as the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) combine as the quasi-regular 137-point (..-cell) 4-polytope, an object worthy of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Why play with the blocks ==

&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;

:: - Carl Jung

&quot;Even the wise cannot see all ends.&quot;

:: - Gandalf

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 16:46, 29 April 2024 (UTC)

== Acknowledgements ==

Coxeter discovered and explored the 11-cell and the 57-cell, leading us up to the ...-cell. He would have inevitably discovered that too, if he had lived even longer, and I am sure that if he had, this article would be just another like the Wikipedia articles I have edited popularizing Coxeter's [[W:Regular Polytopes|Regular Polytopes]] on the web. I see no possible scenario in which I could have conceived the ...-cell myself, built the first model of it, and presented it to Coxeter, even two minutes before he saw it.{{Efn|name=Snelson and Fuller}} Rather, my sight of it comes to me directly from him and others, two decades after him. I might have looked for the 11-point Blind 4-polytope for the rest of my life without finding it, if Tom Ruen had not created a Wikipedia article citing the Blind's discovery of it. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:14, 1 May 2024 (UTC)

I am indebted to [[W:User:Tomruen|Tom Ruen]] for the original illustration of the Coxeter plane of the ...-cell, which he made during pre-publication review of this paper. Further, this research would never have occurred if Ruen had not originated and illustrated a large series of Wikipedia articles on polytopes. The [[W:triacontagon|triacontagon]] article that he illustrated so completely is the beating heart of this subject, and it was the lavishly illustrated Wikipedia articles on regular 4-polytopes that led me to study them in the first place, and attempt to explain them to myself, by contributing to the articles. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 07:17, 4 May 2024 (UTC)

I am deeply indebted to [[W:User:Jgmoxness|J. Gregory Moxness]], first of course for his rendering of Moxness's 60-point (Hull #8) in the 120-cell, and his prior publication of the first image of it captured in the wild, but also personally for his illuminating pre-publication review of this paper, in which he saved me from several egregious errors, about which the less said the better. Second, I am beyond grateful for his splendid transparent renderings of the 11-cell and the ...-cell that he generously made for inclusion in this paper. Moxness and the quaternion graphics software he built are responsible, in two ways, for the gift that we can now see these objects with our own two eyes. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:31, 3 May 2024 (UTC)

==Appendix==

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== References ==
{{Refbegin}}
* {{Citation|title=The Golden Ratio: A Contrary Viewpoint|first=Clement|last=Falbo|journal=The College Mathematics Journal|volume=36|issue=2|date=Mar 2005|pp=123-134|publisher=Mathematical Association of America|jstor=30044835|url=https://www.researchgate.net/publication/247892441_The_Golden_Ratio-A_Contrary_Viewpoint}}


{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="220853" sha1="294puhc8kx2fi7ywuignbsl1cyvqz15" xml:space="preserve">== Bitruncating the 5-cells ==

{{Efn|There are 30 and only 30 distinct chordal distances between vertices in the 120-cell. The Petrie polygon of the 120-cell is a skew {30} triacontagon. The 120-cell's edge is the {30/1} chord. The 5-cell edge chord is the {30/8} chord, which connects two vertices that are 8 120-cell edges apart on the 30-edge Petrie polygon. In the regular 5-cell the characteristic isoclinic rotation in the edge planes which takes the 5 vertices to each other has period 15, and takes the 30 vertices of a compound of 6 disjoint regular 5-cells (called the [[W:Bitruncated 5-cell|bitruncated 5-cell]]) to each other. A regular 5-cell's 10 edges lie in 10 distinct {12} central planes, and the 6 5-cells of the compound share the same set of 10 central planes. The invariant planes of rotation are the 10 {12} central planes containing the 104.5° 5-cell edges, and the 41.1° isocline chords of the rotation are bitruncated 5-cell edges. In each step of the rotation a vertex moves 41.1° over a bitruncated 5-cell edge to a vertex position of a disjoint 5-cell in the compound. In the complementary characteristic isoclinic rotation of the bitruncated 5-cell, the invariant planes of rotation are 15 central planes containing the 41.1° bitruncated 5-cell edges (these are not {12} central planes), and the 75.5° isocline chords of the rotation lie in the 10 {12} central planes containing all the 104.5° 5-cell edges; they are the 180° complements of the 5-cell edge chords, and form great rectangles with them.|name=characteristic 5-cell rotation}}

It is difficult to illustrate the 30-vertex bitruncated 5-cell informatively by projecting it to its 3-dimensional polyhedral shadow. One projection of this hull is a regular icosahedron, but that is deceptive because there are actually 60 edges, not 30, and the 20 triangle faces only meet at their vertices, not at their edges. The triangle faces meet the hexagon faces at their edges. Another projection is an icosidodecahedron, which has 30 vertices but is also misleading because its pentagon faces are not faces of the 4-polytope and its cells at all. The hexagons which ''are'' faces are not visible in either of these projections.

== Truncations of the 5-cells ==
...

* Rectifying the radius 2{{radic|2}} 5-cell of edge 2{{radic|5}} creates the radius {{radic|2}} rectified 5-cell of edge {{radic|5}}.

* Rectifying the radius 2 5-cell of edge 2{{radic|5/2}} creates the unit-radius rectified 5-cell of edge {{radic|5/2}}.

* Rectifying the radius {{radic|2}} 5-cell of edge {{radic|5}} creates the radius {{radic|1/2}} rectified 5-cell of edge {{radic|5}} / 2.

* Rectifying the unit-radius 5-cell of edge {{radic|5/2}} creates the radius 1/2 rectified 5-cell of edge {{radic|5/2}} / 2.


...

== Questions regarding Blind's 11-point augmented rectified 5-cell ==
What happens when we rectify the 5-cells in the 120-cell? Rectifying the 5-cell creates a 10-point (10-cell) semi-regular 4-polytope with 30 equilateral triangle faces, known as the [[W:Rectified 5-cell|rectified 5-cell]]. If we truncate the 600 vertices of a 120-cell this way, we create new vertices at the mid-edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. In a 120-cell of radius {{radic|2}}, the smaller 4-polytope thus created will have a diameter of {{radic|3}}. The 120-cell contains 120 5-cells and many pairings of 5-cells inscribed in opposing positions, with a pair of antipodal vertices, edges, faces, or cells. Here we are concerned with pairs of 5-cells with a pair of antipodal edges. The 5-cell mid-edge to 5-cell mid-edge diameter of the smaller 3-sphere is the {{radic|3}} chord of 75.5°, the 180° complement of the {{radic|5}} 5-cell edge chord of 104.5°. These two chords lie in the 200 irregular {12} dodecagon central planes, where they form three great rectangles.

How many vertices does the smaller 4-polytope have? That is, how many 5-cell edge intersections occur in the 120-cell? The answer depends on whether or not the 120-cell contains pairs of 5-cells in dual position, the edges of which intersect. The intersection of a dual-position 5-cell pair is known to be a 30-point (10-cell) [[W:Bitruncated 5-cell|bitruncated 5-cell]], with 20 triangle faces and 20 hexagon faces.

If 5-cells in dual position exist in the 120-cell, do the 5-cell edges intersect orthogonally? If so, by symmetry do ''four'' 5-cells lie in dual position in the 120-cell (six pairs of 5-cells in dual position), with four orthogonal 5-cell edges at each intersection point, where none of the four orthogonal lines is an axis intersecting the center of the 120-cell?

We can see from the irregular {12} central plane chord diagram that 5-cell edges in different planes will not intersect each other at their mid-points; rather, each 104.5° 5-cell edge has two such points of intersection, 60° apart. The distance between them on the 5-cell edge is a 24-cell edge of the smaller 120-cell. Is a 24-cell edge of the smaller 120-cell colinear with the center portion of a 5-cell edge of the larger 120-cell? Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges meet in cubic vertex figures. The 24-cell's edges also meet in a cubic vertex figure. Are the {{radic|5}} 5-cell edges of the larger 120-cell the {{radic|1}} 24-cell edges of the smaller 120-cell, extended at both ends?

If four 5-cells lie in dual position with their 10 edges intersecting orthogonally at one of their two points of intersection, must the same four 5-cells also intersect orthogonally at the other point of intersection in each edge? Are the four disjoint 5-cells in dual-position Clifford parallel and one 24-cell edge apart? If the 120 5-cells lie in orthogonal dual position in groups of four, the 1200 5-cell edges with two intersection points each are reduced to 600 distinct vertices, making the smaller 4-polytope a smaller 120-cell.

Is the smaller 4-polytope whose vertices are the 5-cell mid-edge intersections also a 600-point 120-cell, so that the 120-cell is concentric to a smaller-radius instance of itself?{{Efn|The self-dual 24-cell is concentric to a smaller instance of itself of half its radius, which constitutes its [[24-cell#Relationships among interior polytopes|common core]], shared with its inscribed 8-cells and 16-cells.}} In that case the 10-point (10-cell) rectified 5-cell is found in the 120-cell, and its edges are chords joining the 600 vertices of the 120-cell: are they 24-cell edge chords? Where exactly in the 120-cell do we find the tetrahedral and octahedral cells of the rectified 5-cell? In particular, are the rectified 5-cell octahedral cells also 24-cell octahedral cells? {{Efn|The rectified 5-cell is said to be the vertex figure of the [[W:5-demicube|5-demihypercube]], the 5-dimensional analog of the 4-demihypercube (16-cell) and the 3-demihypercube (tetrahedron).}}

What is the geometric relationship between these two 10-cell 4-polytopes, the 30-point bitruncated 5-cell and the 10-point rectified 5-cell? Are their triangle faces congruent? The cells of the [[W:Bitruncated 5-cell|bitruncated 5-cell]] are [[W:Truncated tetrahedron|truncated tetrahedra]], with 4 triangle faces and 4 hexagon faces. Where in the 120-cell do we find these truncated tetrahedra? Are their triangle faces 24-cell faces? What are their hexagon faces?

Does the 120-cell also contain pairs of 24-cells in dual position, which intersect at their mid-edges? The mid-edge intersection points of 24-cells in dual position would lie on a third 3-sphere, intermediate in diameter between the original 120-cell and the smaller 120-cell. If this intermediate-size concentric 4-polytope exists within the 120-cell, how many vertices does it have? Do the 24-cell edges (which are also tesseract edges) intersect at their mid-points, and are they orthogonal? If so, by symmetry do ''three'' 24-cells lie in dual position in the 120-cell (three pairs of 24-cells in dual position), with three orthogonal intersecting edges, where the fourth orthogonal line at each intersection point is a 24-cell mid-edge to mid-edge diameter axis intersecting the center of the 120-cell?

The geometers who categorized the [[W:Blind polytope|Blind polytopes]] in 1979 discovered a semi-regular 11-point convex 4-polytope with 42 equilateral triangle faces and 36 edges, an [[W:Rectified 5-cell#Other forms|augmented rectified 5-cell]]. It is constructed by adding one point to the 10-point (10-cell) rectified 5-cell, raising an octahedral pyramid on one (actually on each) of the 10-cell's 5 octahedral facets, creating another semi-regular convex 4-polytope with 13 tetrahedral and 4 octahedral facets: a 17-cell. If the 10-point rectified 5-cell is found in the 120-cell, is this 11-point (17-cell) 4-polytope that is made from it also found in the 120-cell? If so, where exactly, and in what incidence? What is its relation to the tetrahedral and octahedral cells of the rectified 5-cell, and to the truncated tetrahedral cells of the bitruncated 5-cell? What is its relation to the 120-cell's 5-cell edge {30/11} chord, and the isoclinic rotation of period 30 over that chord, which takes hemi-icosahedral rhombicosidodecahedron cells of the 11-cell to each other? Is the 11-point Blind 4-polytope the concrete realization of the abstract 11-point 11-cell?

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:04, 12 April 2025 (UTC)

== Questions regarding Blind's 11-point augmented rectified 5-cell ==
The rectified 5-cell is a Blind 4-polytope, a convex 4-polytope whose cells are all regular polyhedra. The geometers who categorized the [[W:Blind polytope|Blind polytopes]] in 1979 also discovered an 11-point Blind 4-polytope with 36 edges and 42 equilateral triangle faces, the [[W:Rectified 5-cell#Other forms|augmented rectified 5-cell]]. It is constructed by adding one point to the 10-point (10-cell) [[w:Rectified_5-cell|rectified 5-cell]], raising an octahedral pyramid on one of the 10-cell's 5 octahedral facets, creating a 4-polytope with 13 tetrahedral and 4 octahedral facets: a 17-cell. 

Augmenting the rectified 5-cell in this way, with an 11th vertex above one the 5 octahedral cells, adds 6 identical edges, 12 identical triangle faces, and 7 identical tetrahedral cells. The 11th vertex is located above the center of the octahedral cell, displaced radially out of the octahedron's hyperplane. The octahedral pyramid thus formed is exactly filled by 8 regular tetrahedral cells, which meet at its center. It is similar to the 600-cell vertex, an icosahedral pyramid exactly filled by 20 regular tetrahedral cells which meet at its center. 

There are five octahedral cells, so there are five ways to augment a rectified 5-cell in this way. If we augment all five octahedral cells, the resulting 15-point Blind polytope has 60 edges, 90 triangle faces, and 45 tetrahedral cells. 

The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. The tetrahedral-octahedral honeycomb has a cuboctahedron vertex figure, so in the rectified 5-cell, 12 edges radiate in 3-space from each vertex. The infinite tetrahedral-octahedral honeycomb's vertex figure is a radially equilateral cuboctahedron, because its cell radius equals its edge length. But the rectified 5-cell is not radially equilateral: its radius and edge length are unequal, and the 12 radii of its cuboctahedral vertex figure do not meet at 60° angles. ....

Although the 10-point rectified 5-cell is not found in the 120-cell, is this 11-point (17-cell) 4-polytope that is made from it found in the 120-cell? Where exactly is it found, and in what incidence? What is its relation to the tetrahedral and octahedral cells of the [[w:Rectified_5-cell|rectified 5-cell]], and to the truncated tetrahedral cells of the [[w:Truncated_5-cell|truncated 5-cell]] and [[w:Bitruncated_5-cell|bitruncated 5-cell]]? 

What is its relation to the 120-cell's 5-cell edge {30/11} chord, and the isoclinic rotation of period 30 over that chord, which takes hemi-icosahedral rhombicosidodecahedron cells of the 11-cell to each other?

Is the 11-point Blind 4-polytope a realization of the abstract 11-point 11-cell?

== 11 ==
[[W:11 (number)#Mathematics|The number eleven]] is:

: The third superprime number, occupying a prime position in the sequence of primes:

:: (3, 5, 11, 17, ...) in ordinal prime positions (2nd, 3rd, 5th, 7th, ...).

: The twin prime (2 apart) of 7:

:: &lt;math&gt;7 + 2 = 11&lt;/math&gt;

: Forms a sexy pair (6 apart) with 5 and 17 (a sexy threesome?):

:: &lt;math&gt;5 + 6 = 11&lt;/math&gt;

:: &lt;math&gt;11 + 6 = 17&lt;/math&gt;

: At the center of the series of known [[W:Brocard's problem#Brown numbers|Brown numbers]]:

:: (4,5), (5,11), (7,71)

: which are pairs satisfying the Brocard &lt;math&gt;n!+1 = m^2&lt;/math&gt; equalities:

:: &lt;math&gt;4! + 1 = 5^2 = 25&lt;/math&gt; {{color|red|&lt;-- 24-cells}}

:: &lt;math&gt;5! + 1 = 11^2 = 121&lt;/math&gt; {{color|red|&lt;-- 11-cells}}

:: &lt;math&gt;7! + 1 = 71^2 = 5041&lt;/math&gt;

: The rows of [[W:Pascal's triangle|Pascal's triangle]] are the powers of 11:

:: The rows of Pascal's triangle are the base-10 place-value digits of {{red|&lt;math&gt;\color{red}11^n&lt;/math&gt; in row &lt;math&gt;\color{red}n+1&lt;/math&gt;}}. (In any base number system, the place-value digit composition of each row is divisible by 11.)

:: The rows of Pascal's triangle are the configuration incidence counts of the {{red|&lt;math&gt;\color{red}n&lt;/math&gt;-simplex in row &lt;math&gt;\color{red}n+2&lt;/math&gt;}}. If the counts in each row are assigned {{red|&lt;math&gt;\color{red}+&lt;/math&gt; and &lt;math&gt;\color{red}-&lt;/math&gt; signs for even and odd dimensionality}} elements, the counts in each row sum to 0.

== 137 ==
[[W:137 (number)#Mathematics|The number 137]] is:

: The sum / difference of factorials:

:: &lt;math&gt;137 = 5! + 4! - 3! - 2! + 1!&lt;/math&gt; {{color|red|&lt;-- &lt;math&gt;\color{red}+&lt;/math&gt; and &lt;math&gt;\color{red}-&lt;/math&gt; signs (of dimensionalities?)}}

: A [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form &lt;math&gt;4n+1&lt;/math&gt;, where &lt;math&gt;n=34&lt;/math&gt;:

:: &lt;math&gt;137=4\times 34+1&lt;/math&gt;

: which is the sum of two squares:

:: &lt;math&gt;11^{2}+4^{2} = (121+16)&lt;/math&gt; {{color|red|&lt;-- rotations? }}

: A combination of three terms:

:: &lt;math&gt;4^{3}+3^{4}-2^{3} = (64+81-8)&lt;/math&gt;

: the cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link).

== The 137-point ==

The [[W:Electron|Great Celebrity]] arrives at the party, famous for guarding the mystery of her affiliations.{{Sfn|Mac Gregor 137|2017|loc=''137 and the equation E&lt;sub&gt;w&lt;/sub&gt; + E&lt;sub&gt;z&lt;/sub&gt; = E&lt;sub&gt;top&lt;/sub&gt;''}} She was invited, of course, and we hoped she would put in an appearance, but never dreamed she would arrive so soon. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:17, 1 May 2024 (UTC)

== Four dimensions provide four distinct kinds of stickiness ==

&lt;blockquote&gt;All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.&lt;/blockquote&gt;
Consider these two distinct types of glue. Perhaps the strong force (in physics), which acts over a short distance, is a rubber joint gluing several 1-dimensional polytopes together at a point, and each weaker force (including electromagnetism and gravity), which acts over a longer distance, glues ''n''-dimensional polytopes together (where ''n'' inversely characterizes the strength of the force).
[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 20:52, 17 December 2024 (UTC)

== Chords ==
{| class=&quot;wikitable&quot;
!colspan=4|Central planes of the 120-cell
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
|align=center|{12}&lt;sub&gt;1&lt;/sub&gt;
|align=center|{10}
|align=center|{4}
|-
!style=&quot;white-space: nowrap;&quot;|Edge chords
|align=center|15.5° + 44.5° = 60°
|align=center|36°
|align=center|90°

|- 
!style=&quot;white-space: nowrap;&quot;|Incidence
|align=center|200
|align=center|720
|align=center|2250
|-
!style=&quot;white-space: nowrap;&quot;|
|[[File:Great (12) chords of radius √2.png|250px]]
|[[File:Great (10) chords of radius √2.png|250px]]
|[[File:Regular_star_polygon_30-7.svg|250px]]
|}

== Regular non-convex 4-polytopes and the 5/2 chord ==

There are 16 regular 4-polytopes, and all 16 are inscribed in the 120-cell. In the 6 convex regular 4-polytopes the symbols &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt; are integers: 3, 4, or 5. In the 10 non-convex regular 4-polytopes at least one of those symbols is the rational number 5/2, which we see realized as the #8 chord. That distinctive chord is the edge chord of the regular 5-cell and the 11-cell. It also occurs as an interior chord in the 120-cell and in non-convex regular 4-polytopes, but it does not occur at all in the other 4 regular convex 4-polytopes: the 16-cell, 8-cell, 24-cell or 600-cell.

The #8 5/2 chord does not form a regular flat polygon in a central plane. It forms triangular face planes (in the 5-cell and 11-cell), but those triangles do not lie in central planes. The #8 chord forms ''skew'' {5/2} pentagrams in the 5-cell, with each edge of the pentagram lying in a distinct central plane. There are no ''flat'' {5/2} pentagrams of #8 chords anywhere in the regular 4-polytopes. Some of the non-convex regular 4-polytopes do have flat {5/2} pentagram faces, but their edges are not #8 chords: they are #9 chords of length &lt;math&gt;\phi&lt;/math&gt;. In the regular 4-polytopes, flat #8 pentagrams are illusory; they do not actually occur. Where we may think we see them, that is only a projection (flattening) to a 2-dimensional or 3-dimensional perspective view of a skew pentagram, which is actually a 4-dimensional object. It winds spread-out through all 4 dimensions, twice around the 3-sphere ''without self-intersecting'', before closing its circle.

== Alicia Boole Stott's original formulation of dimensional analogy ==
[[File:The Roots of Three.png|thumb|400px|Alicia Boole Stott's 1910 paper defined operations of expansion and contraction on regular polytopes, beginning with expansion of the 2-simplex equilateral triangle to a regular hexagon.]]
...

If we apply the Boole Stott expansion and contraction operations more generally, by relaxing the requirement that only a single edge length be involved, we can also describe the relationship between polytopes of different edge lengths as instances of expansion or contraction. For example, a unit-edge-length 24-cell can be reached as an expansion of a 16-cell of edge length {{radic|2}}, where the expansion distance between edges is also {{radic|2}}. In 3-sphere space this expansion can be performed in two chiral ways: left- or right-handed. Three concentric 16-cells result, inscribed in a {{radic|1}} edge-length 24-cell, if the expansion is done both ways. Corresponding pairs of their vertices are {{radic|3}} apart, corresponding pairs of their edges are {{radic|2}} apart, corresponding pairs of their cell centers are {{radic|1}} apart, and the 16-cell centers are {{radic|0}} apart.

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:12, 13 July 2024 (UTC)

== 5-cell ==
The 5-cell's &lt;math&gt;A_5&lt;/math&gt; symmetry group is simply the alternating group of 5 elements. &lt;math&gt;A_5&lt;/math&gt; is exceptional in that it is not only the symmetry group of the 4-simplex, it is also the symmetry group of the icosahedron and dodecahedron 3-polytopes.

&lt;blockquote&gt;The rotational symmetry group of the ''n''-simplex (not permitting reflections) is always the alternating group &lt;math&gt;A_n&lt;/math&gt;. When &lt;math&gt;n = 4&lt;/math&gt;, this coincides with the rotational symmetry group of the icosahedron and dodecahedron, which is also &lt;math&gt;A_5&lt;/math&gt;. (I believe this is the only &quot;exceptional isomorphism&quot; between symmetry groups of regular polytopes in any dimension, i.e. one besides the isomorphism between dual polytopes.)

Obviously one can prove this fact by showing that each group happens to be the alternating group on 5 elements, but I am curious whether there is a geometric approach to exhibiting the isomorphism - some natural bijection between rotations of the 4-simplex and rotations of the icosahedron which shows they exhibit the same multiplicative structure, without determining what that structure is. 

- Question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com], 2021&lt;/blockquote&gt;

There is some exceptional kind of inter-dimensional duality between the icosahedron and the 5-cell because of their common &lt;math&gt;A_5&lt;/math&gt; symmetry.

An icosahedron can be inscribed in a tetrahedron such that four icosahedron faces lie in the tetrahedron face planes. An author answering the above question on math.stackechange.com circumscribes five rotated regular tetrahedra around an icosahedron, with their 20 faces in the 20 face planes of the icosahedron, in order to demonstrate that the 5-cell and icosahedron belong to the same &lt;math&gt;A_5&lt;/math&gt; symmetry group. The five rotated tetrahedra are not a regular 5-cell, since they they are not face-bonded to each other, and not a 4-polytope since they all lie in the same hyperplane, but by considering their rotations the author concludes that the 5-cell and the icosahedron have the same &lt;math&gt;A_5&lt;/math&gt; symmetry.

A congruence of this kind between some 12-point (icosahedral) actual 4-polytope and a regular 5-cell must occur in 4-space. We can see where it must share at least one 5-cell face triangle with the 120-cell's rhombicosadodecahedral 8th section beginning with a cell. For each pairing of one of the 60 rhombicosadodecahedral central sections with one of the 20 disjoint 5-cells from which it contains a single inscribed face, it must be possible to identify a smaller, concentric 12-point (icosahedral) 4-polytope with one face in the 5-cell's face's plane. Do not this icosahedron's 20 faces lie in the 10 face planes of the same 5-cell, or in the 10 face planes of its completely orthogonal 5-cell? If we project this smaller icosahedron outward onto the 120-cell, it identifies 12 of the 120-cell vertices that form a regular icosahedron. They are also 12 of the rhombicosadodecahedron's 60 vertices, one from each of its 12 pentagon faces.

== Shlafli's criterion for the regular polytopes ==

== Heptad workspace architecture ==

[[File:Workspace architecture.png|thumb|Heptad round conference tables, hexad cubicals, tetrad corridors, and disjoint pentad digon structural pillars that pass through the tetrad voids between the hexad cubicals.]]
Since long before covid I have worked at home in my study, a square outbuilding that has one concave corner. I no longer go in to an office to work, but my ideal imaginary office cubicals are irregular hexagons abutting a square closet or structural pillar, and opening onto a shared work area with a round table. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 13:48, 4 May 2024 (UTC)

== Pentads and hexads together ==
In all three building cycles, in each step where 4-polytope building blocks are joined together to make a larger 4-polytope, an expansion of some 3-polytope embedded at the center of the 3-sphere is taking place, in which the 3-polytope's points are split into 2, 3, 4, or 5 points each and moved apart uniformly, creating a dimensionally analogous 4-polytope. Alicia Boole Stott invented these uniform expansion operations, and explored various distinct forms of them which turn points into edges, edges into faces, faces into cells, and do combinations of more than one of these things. Her goal was to discover new semi-regular 4-polytopes, by starting with the six regular 4-polytopes which were known and performing expansion and contraction operations on them. She began by studying the dimensional analogies between the five regular polyhedra and their regular 4-polytope analogues, and working out the 3D sections of the regular 4D polytopes, but found many more ways to perform expansions and contractions that paired a regular polytope with a semi-regular analogue. There was even a sixth regular 4-polytope which had no regular polyhedron it was an expansion of: the 24-cell. She found the 24-cell was an expansion of its central section, which is a semi-regular 3-polytope: the cuboctahedron. By working in both directions, she found all 15 Archimedean semi-regular 3-polytopes as contractions of regular 4-polytopes, and discovered their 45 Archimedean semi-regular 4-polytope analogues.

Heinz Hopf discovered the general form of these dimensional analogy expansion-contraction operations, which is that every point in the original 3-polytope expands into a linked great circle polygon in the resulting 4-polytope. Each uniform expansion operation determines a distinct dimensional analogy between some 3-polytope Hopf map and 4-polytope Hopf fiber bundle of linked great circle polygon fibers, but every uniform expansion operation turns every point in the 3-polytope into a polygon in a dimensionally analogous 4-polytope, where the polygons are all disjoint but linked together, each passing through all the others. That, it turns out, is the general nature of uniform expansion operations on the 3-sphere. Of course they can be run backwards, as contraction operations, and sometimes a construction is not possible as a straightforward expansion operation (there is no way to get there by expansion alone) but the dimensionally analogous result can be reached by expanding beyond it, and then contracting to it. There is a complexity that arises in doing so, however, because expansion and contraction operations can be paired in two chiral ways (left-handed and right-handed), since they do not commute. Expanding first and then contracting produces a different result than contracting first and then expanding, for the exact same pair of operations on the same object.

== Sequencing the 4-polytopes by complementary chord pairs ==

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=7|Sequencing the 4-polytopes by complementary chord pairs
|-
!
!
!pentads
!hexads
!heptads
!
!
|- style=&quot;background: palegreen;&quot;|
|#1&lt;br&gt;△&lt;br&gt;15.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|
|[[File:120-cell.gif|100px]]&lt;br&gt;600-point (120-cell)
|[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|#14&lt;br&gt;△164.5~°&lt;br&gt;&lt;br&gt;
|- style=&quot;background: seashell;&quot;|
|#2&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;25.2~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:Truncatedtetrahedron.gif|100px]]&lt;br&gt;100 12-point (Lagrange)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|#13&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;154.8~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: yellow;&quot;|
|#3&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;36°&lt;br&gt;{{radic|}}&lt;br&gt;𝝅/5
|[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|#12&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;&lt;br&gt;144°&lt;br&gt;{{radic|}}&lt;br&gt;4𝝅/5
|- style=&quot;background: seashell;&quot;|
|#4&lt;br&gt;△&lt;br&gt;44.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
|
|[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|100px]]&lt;br&gt;120 60-point (Moxness)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;11 137-point (137-cells)
|[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|#11&lt;br&gt;△&lt;br&gt;135.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5&lt;br&gt;△&lt;br&gt;60°&lt;br&gt;{{radic|2}}&lt;br&gt;𝝅/3
|[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
|
|[[File:24-cell.gif|100px]]&lt;br&gt;225 24-point (24-cells)
|
|[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|#10&lt;br&gt;△&lt;br&gt;120°&lt;br&gt;{{radic|6}}&lt;br&gt;2𝝅/3
|- style=&quot;background: yellow;&quot;|
|#6&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;72°&lt;br&gt;{{radic|}}&lt;br&gt;2𝝅/5
|[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
|
|[[File:600-cell.gif|100px]]&lt;br&gt;5 120-point (600-cells)
|
|[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|#9&lt;br&gt;&lt;big&gt;✩&lt;/big&gt;108°&lt;br&gt;{{radic|}}&lt;br&gt;3𝝅/5
|- style=&quot;background: paleturquoise;&quot;|
|#7&lt;br&gt;&lt;big&gt;☐&lt;/big&gt;&lt;br&gt;90°&lt;br&gt;{{radic|4}}&lt;br&gt;𝝅/2
|[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|[[File:5-cell.gif|100px]]&lt;br&gt;120 5-point (5-cells)
|[[File:16-cell.gif|100px]]&lt;br&gt;675 8-point (16-cells)
|[[File:Pentahemicosahedron.png|100px]]&lt;br&gt;120 11-point (11-cells)
|[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|#8&lt;br&gt;△&lt;br&gt;104.5~°&lt;br&gt;{{radic|}}&lt;br&gt;
|-
!
!
!pentads
!hexads
!heptads
!
!
|}

See also the elements, properties and metrics of the sequence of convex 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:33, 23 May 2024 (UTC)

== Progression of the rational numbers as polytopes ==

The rational numbers are described as unit-radius regular 4-polytopes inscribed concentric to the 3-sphere. Except where explicitly described otherwise, polygons are regular great polygons in a central plane, not face polygons; regular polyhedra are inscribed at the center of the 3-sphere, not as cells or vertex figures; and 4-polytopes are one torus of the face-bonded cells of a regular convex 4-polytope, not the complete honeycomb that includes all its cells.

The 0-point {0} has no facets, and no extent. The 1-point {1} unigon has 1 non-facet a {{radic|1}} 0-point, and no 0-extent. The 2-point {2} digon has 2 0-facets each a {{radic|2}} unigon, and 1-extent {{radic|2&lt;sup&gt;2&lt;/sup&gt;/1&lt;sup&gt;4&lt;/sup&gt;}} = 2. The 3-point trigon {3} has 3 1-facets each a {{radic|3}} digon, and 2-extent {{radic|3&lt;sup&gt;3&lt;/sup&gt;/2&lt;sup&gt;4&lt;/sup&gt;}} = {{radic|27/16}} = 1.299. The 4-point quadragon {4} has 4 1-facets each a {{radic|2}} digon and 2-extent {{radic|4&lt;sup&gt;3&lt;/sup&gt;/2&lt;sup&gt;4&lt;/sup&gt;}} = {{radic|64/16}} = 2. It also has 2 diagonals each a unit-radius digon (an axis diameter) whose 1-extent is equal to that 2-extent. Thus it has 6 digons making an irregular flat tetrahedron with 4 non-disjoint right-triangle faces lying in one central plane.

The 4-point tetrahedron has 4 2-facets each a {{radic|5/2}} trigon and 3-extent {{radic|4/3}}; 6 1-facets each a {{radic|..}} digon. The 5-point 5-cell has .... [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:44, 28 May 2024 (UTC)

= A symmetrical arrangement of eleven 11-cells (draft notes by section) =

== 5-cells and hemi-icosahedra in the 11-cell ==

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} &lt;s&gt;The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell.&lt;/s&gt; The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron.

''prior to'' [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:42, 18 December 2024 (UTC)

== The eleventh chord ==

That projection from a {30/10} chord in the stationary 4-dimensional reference frame to a {30/8} chord in the rotating 2-dimensional reference frame, of a moving invariant plane as it tilts sideways in 4-space, is a [[W:Lorentz transformation|Lorentz transformation]]. It is the equation which relates an actual chordal distance within a 4-polytope to the foreshortened chord which it appears to be when seen in perspective view, as a 3-polytope or 2-polytope. The same relation expresses the equivalence (and equal validity) of two different systems of measurement, one in a rotating reference frame and one in its stationary reference frame, even though they yield distinctly different lengths.
[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:10, 7 March 2025 (UTC)

== Compounds in the 120-cell ==

...

=== Building the building blocks themselves ===

...

And here at last with the pentad and hexad orthoschemes we must be able to find formulae describing each polytope that relate &lt;math&gt;pentads^4 = hexads^4&lt;/math&gt;, the two equivalent constructions from root systems for every 4-polytope, even for polytopes which are obviously constructed one way, and not so obviously the other. One possible construction is always by pentad orthoschemes, since it is only necessary to construct the polytope's characteristic 4-orthoscheme, and 4-orthoschemes are pentads themselves. The other is by the hexads of the 16-cell, since 16-cells compound into everything larger. Surely every uniform 4-polytope can be constructed by some function of either of these root systems, however indirect the recipe. The expressions to do so then, with an equal sign between them, make a conservation law defining the 4-polytope, which we may call its physics by Noether's theorem. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 07:11, 26 April 2024 (UTC)

...

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

...see also the 120 right-triangle-faced, 62-point [[W:Disdyakis triacontahedron|disdyakis triacontahedron]]. As the barycentric subdivision of the regular dodecahedron and icosahedron, it contains the decomposition of all the &lt;math&gt;H_4&lt;/math&gt; polytopes to orthoschemes. It has a close relationship to the 60-point [[W:Rhombic triacontahedron|rhombic triacontahedron]]....

=== 120 of them ===
The{{radic|2}}-radius 120-cell has 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles. It contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== What's in the box ===
This is the picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell, including all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks &lt;s&gt;by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures&lt;/s&gt;.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). &lt;s&gt;The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.&lt;/s&gt;{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the torus decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers cited in the Wikipedia article make clear, but his description of its multi-faceted geometry is also the clearest set of instructions available for how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct manner in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a rotation of a distinct 4-polytope on the 3-sphere. The crucial point is that it is the rotation object in 4-space which is the real object, and the map in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they cannot justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a 120-cell cell ring (each fiber in the fiber bundle) is conflated to one distinct point on a 12-point (regular icosahedron) map. The characteristic rotation of the 120-cell which takes dodecahedral cells to each other within their respective rings of 10 dodecahedra is described by this Hopf map. Within the 120-cell (the territory mapped) we find an distinct instance of this map for every instance of the rotation it describes. The 120-cell contains 10 600-cells, each of which has 120 vertices, each with a regular icosahedron vertex figure that is the Hopf map of a distinct rotation.

In like fashion, we can look within the 120-cell for the Hopf maps of the 11-cell's characteristic rotations. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure, in complement to the way Moxness's 60-point rhombicosadodecahedron revealed their internal structure. We have already identified the 11-cell's characteristic rotation, in which the 11 vertices circulate on 11 disjoint {30/11} skew polygrams. The Hopf map of this rotation of the real 11-cell 4-polytope will be some 3-polytope that is an abstraction of it. Of course that abstract map must be the abstract 3-polytope we began with, the hemi-icosahedron, or more precisely, its realization as Moxness's 60-point rhombicosadodecahedron 3-polytope.

The obvious place to look for the 11-cell's Hopf map is near Moxness's 60-point rhombicosadodecahedron; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron and consider it as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that regular dodecahedra form 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. &lt;s&gt;But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere&lt;/s&gt;.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The rational numbers are each a distinct flavor.

Each natural number has its own distinct taste. The first six {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are prime integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is just mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct rational number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and distinctive flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the rational number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. Each distinct #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord is some unique rational number &lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;. The #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f(h)+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons (not all shown here): a compound of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons), plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the #11 chord, shown with a non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram so that it lies parallel to the #8 chord. These two chords actually do lie parallel in the same {12} central planes.]]

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct rational numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity (as we saw [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|above]]), they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

These 15 chords (rational numbers) include the natural numbers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the ratios 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions, as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 9 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly, unless you have Steinbach's formula:
{| style=&quot;background-color:transparent; white-space:nowrap; float:left;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}
The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'' before closing their circuit, a helical geodesic circle or ''isocline'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f(h)&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a rational number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the rational number.

{| class=wikitable style=&quot;white-space:nowrap;&quot;
!colspan=6|Rational number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{3,\tfrac{5}{2}\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct rational number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the rational number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The real 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. &lt;s&gt;The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra.&lt;/s&gt; The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces &lt;s&gt;which lie between three cells&lt;/s&gt;, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

&lt;s&gt;Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|&lt;s&gt;Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves two ways: as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.&lt;/s&gt;|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.&lt;/s&gt;

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-hypercubes) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-hypercubes) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

== The 11-cell's real elements ==

We seek an object in the 120-cell which is the realization of the abstract 11-cell. The real 11-cell has central section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra, while the abstract 11-cell has hemi-icosahedron cells. How are the section 8&lt;sub&gt;3&lt;/sub&gt; rhombicosidodecahedra different than the abstract hemi-icosahedra?

Consider their incidences, previously described: 

* Each 11-cell contains 11 of the 120-cell's 120 completely disjoint 5-cells.
* Each hemi-icosahedron shares a face with 10 distinct 5-cells, and is completely disjoint from 1 of the 5-cells. Each 5-cell shares 1 of its 10 faces with each of 10 hemi-icosahedra, and is completely disjoint from 1 hemi-icosahedron.
* Each rhombicosidodecahedron shares a face with 20 distinct 5-cells, and is completely disjoint from 100 of the 5-cells.{{Efn|Each rhombicosidodecahedron contains only 20 5-cell faces, each from a distinct 5-cell, so it is face-disjoint from 100 of the 5-cells. It contains only 60 5-cell edges, so it is edge-disjoint from those 100 5-cells. It contains only 60 5-cell vertices, so it is vertex-disjoint from those 100 5-cells.}} Each 5-cell shares 1 of its 10 faces with each of 10 rhombicosidodecahedra, and is completely disjoint from 50 rhombicosidodecahedra.
* Each hemi-icosahedron shares its 10 faces (each containing 3 distinct 5-cell edges) with 10 other hemi-icosahedra.
* Each rhombicosidodecahedron shares its 10 {12} central planes (each containing 6 disjoint 5-cell edges) with 10 other rhombicosidodecahedra.

The abstract hemi-icosahedron's 5-cell faces do not actually meet at an edge; each hemi-icosahedron face is actually two 5-cell faces in different places; and 5-cell faces are disjoint in the rhombicosidodecahedra and in the 11-cell. Clearly, the 5-cell edges are not the 11-cell's real edges. The actual 11-cell edge chord is not the 5-cell edge {30/8} chord of length {{radic|5}}, but the longer {30/10) chord of length {{radic|6}}. This [[24-cell#Helical hexagrams and their isoclines|isocline chord of the 24-cell]] is the ''rotation'' e''dge chord'' of the 11-cell.

The 5-cell faces are not the real 11-cell's faces. The true face of the 11-cell is a great triangle of edge length {{radic|6}} (shown in green in the illustration), lying in a {12} central plane. The green great triangle is where two rhombicosidodecahedrons meet face-to-face as cells do. In the six regular convex 4-polytopes, 3-polytope cells meet pairwise at non-central face planes, and form a honeycomb of voluemtrically disjoint cells. Additionally, four of these regular 4-polytopes (all but the two simplest, the 5-cell and 16-cell) are compounds of smaller regular 4-polytopes, which meet pairwise at central planes. The simplest case is the 16-point (8-cell) tesseract, a compound of two 8-point (16-cells), which meet 2 at a {6} hexagon central plane. The 24-point (24-cell) is a non-disjoint compound of three 16-point (8-cell) tesseracts, which meet 2 at a {6} hexagon central plane. The 120-point (600-cell) is a compound of 5 disjoint 24-point 24-cells (ten different ways), which meet 2 at a {6} hexagon central plane. The 600-point (120-cell) is a compound of 5 disjoint 120-point 600-cells, two different ways, which meet 2 at a {12} dodecagon central plane. All these regular convex 4-polytopes are both a honeycomb of volumetrically disjoint 3-polytope cells, and they also form &quot;inside out honeycomb&quot; compounds of smaller 4-polytope &quot;hypercells&quot;, which have their shared &quot;faces&quot; on the inside, in their central rotation planes.

Like these other 4-polytopes, the 11-cell is both an assembly of volumetrically disjoint 3-polytope cells (they are the tetrahedral cells of its 11 5-cells), -- no it is not one assembly the 5-cells are disjoint -- and also a compound of non-disjoint 4-polytope &quot;hypercalls&quot; meeting 2 at a central plane. Each of its 4-polytope &quot;hypercells&quot; is -- what object exactly? -- comprised of selected parts of two disjoint polyhedra in different places: a completely orthogonal pair of 60-point (central section 8&lt;sub&gt;3&lt;/sub&gt;) rhombicosidodecahedrons. Together those selected parts occupy all 4 dimensions, not just a 3-dimensional polyhedral section, and comprise an 11-point, 10-edge, 15-face 4-polytope. -- doesn't it have to be 15-edge, 10-face? -- The 11-cell is the non-disjoint compound of 11 of these 11-point 4-polytope &quot;hypercells&quot;.

The real 11-cell face is a {{Radic|6}} triangle in the {12} central plane parallel to two 5-cell face planes, which lie {{Radic|3}} above and {{Radic|3}} below it. The orthogonal distance between the two 5-cell faces is another {30/10} chord of length {{Radic|6}}, which joins their opposing vertices. That chord is another great triangle edge, in another {12} central plane, but it does not intersect any vertex of the parallel {12} central plane.

We found previously that the abstract hemi-icosahedron face was a {{radic|5}} 5-cell face; then that it was two 5-cell faces in disjoint 5-cells; then that it was really a {{radic|6}} great triangle in a central plane. These findings are not contradictory. The abstract hemi-icosahedron face is realized in a real rhombicosidodecahedron as a rigid object composed of these three parallel triangles. We will visualize that 9-point polyhedron precisely later, but hold the thought that an abstract 11-cell ''face'' is really a ''polyhedron''. This makes sense since the abstract 11-cell ''cell'' is really a ''4-polytope''. Once again, the 11-cell has something to teach us about interdimensional relationships, and the nature of dimensional analogy. -- where am I going with this suspect notion? --

The {12} central plane contains 4 disjoint great triangles (11-cell faces) in distinct rotational orientations, from 4 distinct 11-cells (although to reduce clutter only one of them is shown in green in the illustration). The {12} is the central plane of four distinct 9-point polyhedra, because it contains four distinct great triangles. The two parallel 5-cell face planes also contain triangles (5-cell faces), but these face planes are only {6}s, not {12}s. Each face plane contains only two 5-cell face triangles in opposing rotational orientations, not four triangles, because two of the four distinct 9-point polyhedra share a 5-cell face, the same way two tetrahedral cells of the 5-cell share the face. Pairs of distinct 9-point polyhedra form a 12-point polyhedron with two opposing 5-cell faces, parallel to a hexagon in the central plane between them.{{Efn|There are two regular great hexagons of edge length {{radic|2}} inscribed in each irregular {12} great dodecagon, although to reduce clutter they are not shown explicitly in this illustration. They are the famous [[24-cell#Great hexagons|24-cell great hexagons]]; each 24-cell has 16 of them. Each has two {{radic|6}} [[24-cell#Great triangles|great triangles]] inscribed in it.}} In each of 10 sets of three parallel planes (two 5-cell face planes on either side of a {12} central plane), we find four 9-point polyhedra (four sets of three parallel triangles), paired as two 12-point polyhedra (two sets of two parallel triangles with a parallel hexagon between them). The two opposing 5-cell faces in each {6} face plane belong to two disjoint 5-cells, and also to two distinct rhombicosidodecahedra. The 10 9-point polyhedra are inscribed in a single rhombicosidodecahedron, but the 12-point polyhedra are not.

...

== The 11-cell rotation ==

Eleven rhombicosidodecahedra are pairwise adjacent in two ways: (1) they are in contact, sharing central planes pairwise, similar to the way two cells of a 4-polytope are bound together at a shared face, and (2) they occupy adjacent Clifford parallel rotational positions in various 30-position isoclinic rotations, not in contact at all, but separated at every vertex by the same small distance. For each pair of rhombicosidodecahedra, these two kinds of adjacency occur in different places. There are places where they share a {{Radic|6}} triangle in a central plane, and other places where their two {{Radic|6}} triangles lie nearby, Clifford parallel, in adjacent positions of an isoclinic rotation which takes them to each other. The rotation takes entire rhombicosidodecahedra to each other's positions, in fact it takes all 60 rhombicosidodecahedra at once to each other's positions; an isoclinic rotation moves everything at once, altogether in parallel, in many different orthogonal directions at once. 

This is rather confusing to visualize, but let us try. In this case, the completely orthogonal invariant planes of the isoclinic rotation are pairs of {12} central planes. The 120-cell's 100 completely orthogonal pairs of {12} central planes all rotate at once, by the same displacement angle in each step of the rotation. In this case the angle is 120°, the arc of one edge of a {{Radic|6}} triangle in each central plane. Visualize 800 spherical triangles (or 200 irregular dodecagons, since there are 4 great triangles in each {12} central plane), all rotating like wheels. Now consider that they are 100 completely orthogonal pairs of wheels. Each is not only rotating like a wheel, it is also being rotated sideways like a coin flipping, each vertex moving on a sideways-moving rotating wheel. The actual path through space of each vertex is a closed spiral, a circular helix.

We are visualizing this as a coin rotating like a wheel while it flips sideways in space at the same time, because that is the closest thing to it we have observed in life, but it is more than that. A coin's twisting rotation takes place only in three dimensions: as a wheel in one 2-dimensional plane, and as a flipped coin in another 2-dimensional plane, but those two planes share an axis, because in 3-space there are only 3 orthogonal axes and 3 orthogonal planes through a central point. In 4-space there are 4 orthogonal axes and 6 orthogonal planes through a point (including 3 pairs of [[W:Completely orthogonal|completely orthogonal]] planes which intersect only at the central point, not anywhere in a line). 4-space is potentially very confusing, but the essential thing to understand about its 4 axes is that 4-space is simply roomier. It is more commodious than the 3-space we are confined in, and it has room to spread out in surprising new ways, such as the way two orthogonal circles need not be crammed into sharing an axis, as they must in 3-space. What this means for our visualization process is that we must relax our cramped view of space, and try to see the two completely orthogonal sideways-moving wheels correctly: each occupies a fully independent 2-axis 2-dimensional plane. It is easier to picture their motions ''incorrectly'', the way they would have to happen in 3-space, but a double rotation in 4-space, which we are not in the habit of visualizing correctly because it is unprecedented in our three-dimensional experience, is nonetheless accessible to the human visual imagination, which is a very powerful image processing engine that understands dimensional analogy really well.

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== The 11-cell Hopf fibration ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the torus decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers cited in the Wikipedia article make clear, but his description of its multi-faceted geometry is also the clearest set of instructions available for how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct manner in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a rotation of a discrete 4-polytope on the 3-sphere. The crucial point is that it is the rotation object in 4-space which is the real object, and the map in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they cannot justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a 120-cell ring of dodecahedra (each fiber in the fiber bundle) is conflated to one distinct point on a 12-point (regular icosahedron) map. The characteristic rotation of the 120-cell which takes dodecahedral cells to each other within their respective rings of 10 dodecahedra is described by this Hopf map. Within the 120-cell (the territory mapped) we find an distinct instance of this map for every instance of the rotation it describes. The 120-cell contains 10 600-cells, each of which has 120 vertices, each with a regular icosahedron vertex figure that is the Hopf map of a distinct isoclinic rotation.

In like fashion, we can look within the 120-cell for the Hopf maps of the 11-cell's characteristic rotations. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure, in complement to the way Moxness's Hull #8 rhombicosidodecahedron reveals the cell's internal structure. We have already identified the 11-cell's characteristic rotation, in which the 11 vertices circulate on 11 disjoint {30/11} skew polygrams with {30/11} chord edges. The Hopf map of this rotation of the real 11-cell 4-polytope will be some 3-polytope that is an abstraction of it. Of course that abstract map must be the abstract 3-polytope we began with, the hemi-icosahedron, or more precisely, its realization as Moxness's Hull #8 rhombicosidodecahedron 3-polytope.

The rhombicosidodecahedron map has 60 vertices, and each vertex must ''lift'' to a disjoint great circle polygon of the 11-cell. The great circle polygons of the 11-cell rotation are its rotation edge polygons, which we found are {{Radic|6}} great triangles with {30/10} chord edges. These 60 disjoint great circle polygons must exactly fill the real 11-cell 4-polytope, comprising all its vertices. Thus the real 11-cell has 180 vertices. The 11 abstract vertices circulate on 11 disjoint 30-position {30/11} skew polygram isoclines, each abstract vertex visiting one-sixth {30} of the 180 vertices during the rotation. Each abstract vertex is a conflation of .. real vertices on .. {30/11} isoclines, and is represented at one time or another during the rotation by .. distinct vertices which its rotating real vertices visit. There are 11 discrete Hopf fibrations of the 11-cell, with 60 disjoint great {3} triangle fibers each, and 11 disjoint skew {30/11} triacontagram isocline fibers each. Each fibration corresponds to a distinct left (and right) instance of the characteristic isoclinic rotation.

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=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. &lt;s&gt;The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since&lt;/s&gt; &lt;s&gt;the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy&lt;/s&gt;

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cube). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 4-polytopes, so e.g. the 5-cell edge is {{radic|5}}. Here we give two illustrations of the Jessen's with different metrics: the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius. The correspondence between our 3-space and 4-space metrics is curiously cyclic; in the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, and which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra, in three different rhombicosidodecahedra, and like most polyhedron faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell ''faces'' at all, but central polygons of the cell. Of course they are the {{radic|6}} triangles seen in the irregular {12} dodecagon central planes, which we saw are the actual shared &quot;faces&quot; between adjacent hemi-icosahedron cells. Opposing {{radic|6}} triangles lie in completely orthogonal {12} central planes, where they are inscribed in great hexagons (but not in the same great hexagon of course, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] {12} planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The {{radic|5}} 5-cell faces do not appear in this illustration, only some of their edges do. The Jessen's (the building block we found 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 isoclinic cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being isoclinic rotations of each other. 

...

...removed the following from a footnote to the article, where it refers to the 5-cell edges that meet at 120-cell pentagon faces, because I am not sure it is true; possibly only the 11-cell edges, which meet at triangle faces, can act as the struts of a stable tensegrity structure (since pentagons unlike triangles are not rigid structures)
&lt;blockquote&gt;Consequently the 120-cell can be constructed as an infinitesimally mobile rigid geodesic 3-sphere: a 4-dimensional tensegrity sphere. The 120-cell's 1200 edges need only be tension cables, provided that a disjoint 600 of the 120 5-cells' 1200 edges are included as compression struts, in parallel pairs.|name=tensegrity 120-cell
&lt;/blockquote&gt;

...

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* Wright's architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', &lt;s&gt;an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}&lt;/s&gt;
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-hypercube), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-hypercube]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-hypercubes), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==

The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its 600 instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton honeycomb 4-polytope or a non-disjoint compound 4-polytope, unless we call it both. We should not even firmly ascribe to it a distinct 4-dimensionality, since its 137-point compound contains or induces instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but a central point about them is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} Stott applied her operations to uniform polytopes of only one edge length recursively, starting with the regular polytopes, which did not lead her to the 11-cell, even though it has only one edge length. It is only because she did not have the 11-cell to start with that she did not discover the 137-point (..-cell) by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes, and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries, the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all these things singly, like the regular convex 4-polytopes, it compounds to significantly more.

The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has its realization in Euclidean 4-space as this convex 4-polytope with regular facets and regular triangle faces.

The 11-cell (plural) is subsumed in the 120-cell, just as the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) combine as the quasi-regular 137-point (..-cell) 4-polytope, an object worthy of further study.

The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Why play with the blocks ==

&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;

:: - Carl Jung

&quot;Even the wise cannot see all ends.&quot;

:: - Gandalf

[[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 16:46, 29 April 2024 (UTC)

== Acknowledgements ==

Coxeter discovered and explored the 11-cell and the 57-cell, leading us up to the ...-cell. He would have inevitably discovered that too, if he had lived even longer, and I am sure that if he had, this article would be just another like the Wikipedia articles I have edited popularizing Coxeter's [[W:Regular Polytopes|Regular Polytopes]] on the web. I see no possible scenario in which I could have conceived the ...-cell myself, built the first model of it, and presented it to Coxeter, even two minutes before he saw it.{{Efn|name=Snelson and Fuller}} Rather, my sight of it comes to me directly from him and others, two decades after him. I might have looked for the 11-point Blind 4-polytope for the rest of my life without finding it, if Tom Ruen had not created a Wikipedia article citing the Blind's discovery of it. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 17:14, 1 May 2024 (UTC)

I am indebted to [[W:User:Tomruen|Tom Ruen]] for the original illustration of the Coxeter plane of the ...-cell, which he made during pre-publication review of this paper. Further, this research would never have occurred if Ruen had not originated and illustrated a large series of Wikipedia articles on polytopes. The [[W:triacontagon|triacontagon]] article that he illustrated so completely is the beating heart of this subject, and it was the lavishly illustrated Wikipedia articles on regular 4-polytopes that led me to study them in the first place, and attempt to explain them to myself, by contributing to the articles. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 07:17, 4 May 2024 (UTC)

I am deeply indebted to [[W:User:Jgmoxness|J. Gregory Moxness]], first of course for his rendering of Moxness's 60-point (Hull #8) in the 120-cell, and his prior publication of the first image of it captured in the wild, but also personally for his illuminating pre-publication review of this paper, in which he saved me from several egregious errors, about which the less said the better. Second, I am beyond grateful for his splendid transparent renderings of the 11-cell and the ...-cell that he generously made for inclusion in this paper. Moxness and the quaternion graphics software he built are responsible, in two ways, for the gift that we can now see these objects with our own two eyes. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:31, 3 May 2024 (UTC)

==Appendix==

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}}

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}}

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== References ==
{{Refbegin}}
* {{Citation|title=The Golden Ratio: A Contrary Viewpoint|first=Clement|last=Falbo|journal=The College Mathematics Journal|volume=36|issue=2|date=Mar 2005|pp=123-134|publisher=Mathematical Association of America|jstor=30044835|url=https://www.researchgate.net/publication/247892441_The_Golden_Ratio-A_Contrary_Viewpoint}}


{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>294puhc8kx2fi7ywuignbsl1cyvqz15</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
    <id>326765</id>
    <revision>
      <id>2805804</id>
      <parentid>2805712</parentid>
      <timestamp>2026-04-21T18:03:12Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */ better description of double rotation</comment>
      <origin>2805804</origin>
      <model>wikitext</model>
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      <text bytes="31722" sha1="719jofdw6n0tic58558v7lq43taydlh" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]].

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

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

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>719jofdw6n0tic58558v7lq43taydlh</sha1>
    </revision>
    <revision>
      <id>2805806</id>
      <parentid>2805804</parentid>
      <timestamp>2026-04-21T18:13:33Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2805806</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31781" sha1="n169v3a3jdl0megs2fcxqr6momlc8rw" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

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

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>n169v3a3jdl0megs2fcxqr6momlc8rw</sha1>
    </revision>
    <revision>
      <id>2805807</id>
      <parentid>2805806</parentid>
      <timestamp>2026-04-21T18:34:21Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2805807</origin>
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      <text bytes="31792" sha1="rrlr03ma5lkf0b5pcjr9f9jp1cpuwqp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

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

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>rrlr03ma5lkf0b5pcjr9f9jp1cpuwqp</sha1>
    </revision>
    <revision>
      <id>2805808</id>
      <parentid>2805807</parentid>
      <timestamp>2026-04-21T18:37:27Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2805808</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31817" sha1="obl8pkg6ssb2057zc0ak2w7yppfd18e" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can also be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>obl8pkg6ssb2057zc0ak2w7yppfd18e</sha1>
    </revision>
    <revision>
      <id>2805826</id>
      <parentid>2805808</parentid>
      <timestamp>2026-04-21T21:14:10Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Conclusions */</comment>
      <origin>2805826</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31812" sha1="ou1u2w8oi5ky265gvbqfqtubk8nkb3s" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ou1u2w8oi5ky265gvbqfqtubk8nkb3s</sha1>
    </revision>
    <revision>
      <id>2805829</id>
      <parentid>2805826</parentid>
      <timestamp>2026-04-21T23:02:34Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="31816" sha1="tg4uevd7is49u97r841ii1ak6cf7jup" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>tg4uevd7is49u97r841ii1ak6cf7jup</sha1>
    </revision>
    <revision>
      <id>2805832</id>
      <parentid>2805829</parentid>
      <timestamp>2026-04-21T23:21:58Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2805832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36910" sha1="jx46wtqp89dyfgw70p9tt6gm8clwidm" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>jx46wtqp89dyfgw70p9tt6gm8clwidm</sha1>
    </revision>
    <revision>
      <id>2805833</id>
      <parentid>2805832</parentid>
      <timestamp>2026-04-21T23:22:51Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Introduction */</comment>
      <origin>2805833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37492" sha1="bhrtqu4x4zwrj2vm607zjoksdu8acj5" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>bhrtqu4x4zwrj2vm607zjoksdu8acj5</sha1>
    </revision>
    <revision>
      <id>2805834</id>
      <parentid>2805833</parentid>
      <timestamp>2026-04-21T23:24:41Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="37492" sha1="s2axkcocy39pnyj3yhsiav8vqslydh2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== Thirty distinguished distances ==
[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>s2axkcocy39pnyj3yhsiav8vqslydh2</sha1>
    </revision>
    <revision>
      <id>2805835</id>
      <parentid>2805834</parentid>
      <timestamp>2026-04-21T23:28:54Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Thirty distinguished distances */</comment>
      <origin>2805835</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37512" sha1="o9l5p99pg6sd7z9e0v1wstqxx0isg89" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]

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

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|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>o9l5p99pg6sd7z9e0v1wstqxx0isg89</sha1>
    </revision>
    <revision>
      <id>2805836</id>
      <parentid>2805835</parentid>
      <timestamp>2026-04-21T23:30:14Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="37879" sha1="0habqpir912jljchzn7c40tzmxxlahc" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>0habqpir912jljchzn7c40tzmxxlahc</sha1>
    </revision>
    <revision>
      <id>2805837</id>
      <parentid>2805836</parentid>
      <timestamp>2026-04-22T00:04:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
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      <text bytes="38397" sha1="k5692nketgq12f0mi21yt6rpkrv1p75" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]]. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>k5692nketgq12f0mi21yt6rpkrv1p75</sha1>
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    <revision>
      <id>2805838</id>
      <parentid>2805837</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="38403" sha1="nmfa1y34y0ab3z3nu5lrukegyadow54" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]]. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>nmfa1y34y0ab3z3nu5lrukegyadow54</sha1>
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    <revision>
      <id>2805839</id>
      <parentid>2805838</parentid>
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      <text bytes="38563" sha1="f94z24isxzxiqxxdvkucf6wd6e54wf0" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>f94z24isxzxiqxxdvkucf6wd6e54wf0</sha1>
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    <revision>
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        <username>Dc.samizdat</username>
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      <text bytes="38723" sha1="c5g8msvooughumn0iaemk9ejezcqij8" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular polytopes.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
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!rowspan=2|arc
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!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>c5g8msvooughumn0iaemk9ejezcqij8</sha1>
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    <revision>
      <id>2805841</id>
      <parentid>2805840</parentid>
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      <text bytes="38973" sha1="15vybp327gggmb7dnkl748r7km63i5p" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle and nonagon {9} faces.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>15vybp327gggmb7dnkl748r7km63i5p</sha1>
    </revision>
    <revision>
      <id>2805842</id>
      <parentid>2805841</parentid>
      <timestamp>2026-04-22T00:47:55Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
      <origin>2805842</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="39054" sha1="r9y3s0npk3rm0n0odgtfqv5btcif1jf" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle {3} and nonagon {9} faces.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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!rowspan=2|arc
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!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
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|-
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|-
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|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>r9y3s0npk3rm0n0odgtfqv5btcif1jf</sha1>
    </revision>
    <revision>
      <id>2805843</id>
      <parentid>2805842</parentid>
      <timestamp>2026-04-22T00:51:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="39048" sha1="tfsiaa24k9uadsd7ab2u8uq7l8hhosl" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle {3} and nonagon {9} faces.

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>tfsiaa24k9uadsd7ab2u8uq7l8hhosl</sha1>
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    <revision>
      <id>2805844</id>
      <parentid>2805843</parentid>
      <timestamp>2026-04-22T00:55:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="39018" sha1="h9wazup31ztxiu4v6u8qpdxdsfpnp4z" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle {3} and nonagon {9} faces.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>h9wazup31ztxiu4v6u8qpdxdsfpnp4z</sha1>
    </revision>
    <revision>
      <id>2805845</id>
      <parentid>2805844</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="39599" sha1="sty7o69mc45n917y1iao1vwgqc4smbg" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|A projection to three dimensions of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]
[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle {3} and nonagon {9} faces.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>sty7o69mc45n917y1iao1vwgqc4smbg</sha1>
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    <revision>
      <id>2805846</id>
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      <text bytes="39638" sha1="g72440ibgctlcfgo367mbmldbtog3is" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]
[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As one example, there is a section of the 120-cell which is an irregular 48-point polyhedron with triangle {3} and nonagon {9} faces.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>g72440ibgctlcfgo367mbmldbtog3is</sha1>
    </revision>
    <revision>
      <id>2805856</id>
      <parentid>2805846</parentid>
      <timestamp>2026-04-22T02:34:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
      <origin>2805856</origin>
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      <format>text/x-wiki</format>
      <text bytes="39891" sha1="rvrsbef9yqpdm4bmyd77wbxovih1hn0" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} Although the 120-cell has 30 distinct chord lengths, in this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is transluscent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]
[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Visualizing the interior of the 120-cell is a challenge because of its complexity. Most renderings of the 120-cell only illustrate its outer surface of face-bonded dodecahedral cells, which hides all its interior chords and inner surfaces.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>rvrsbef9yqpdm4bmyd77wbxovih1hn0</sha1>
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    <revision>
      <id>2805872</id>
      <parentid>2805856</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="40354" sha1="ddss2gzc478y46cwzlaqcgxejbh5mmq" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
[[File:120-cell.gif|thumb|360px|Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]
[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. ]]

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2805874</id>
      <parentid>2805872</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="40391" sha1="mkv2c3li5tqmksvzqprhijsdmfn227o" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

== The 120-cell ==
{| class=&quot;wikitable floatright&quot;
|[[File:120-cell.gif|thumb|360px|Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.]]
|[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. ]]
|}

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>mkv2c3li5tqmksvzqprhijsdmfn227o</sha1>
    </revision>
    <revision>
      <id>2805875</id>
      <parentid>2805874</parentid>
      <timestamp>2026-04-22T06:04:44Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
      <origin>2805875</origin>
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      <text bytes="40373" sha1="jchuhfoxe61bemy9q2jpjjqdqdnpmt2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>jchuhfoxe61bemy9q2jpjjqdqdnpmt2</sha1>
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    <revision>
      <id>2805876</id>
      <parentid>2805875</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="40382" sha1="sly4vbt047zoe1xzpg1nq2v29b2fjry" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>sly4vbt047zoe1xzpg1nq2v29b2fjry</sha1>
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    <revision>
      <id>2805877</id>
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      <text bytes="40384" sha1="qqcvvtla342101kcs3rhv8nb6kfw45r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>qqcvvtla342101kcs3rhv8nb6kfw45r</sha1>
    </revision>
    <revision>
      <id>2805878</id>
      <parentid>2805877</parentid>
      <timestamp>2026-04-22T06:08:37Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
      <origin>2805878</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40396" sha1="58b5azrzuvwwu6pko01xdr8xujof0rg" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>58b5azrzuvwwu6pko01xdr8xujof0rg</sha1>
    </revision>
    <revision>
      <id>2805880</id>
      <parentid>2805878</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="40389" sha1="84ydjb1otz0dhi6u6toabsnye6dld2r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

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

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

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

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>84ydjb1otz0dhi6u6toabsnye6dld2r</sha1>
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    <revision>
      <id>2805881</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="40392" sha1="2erhjxw9ttxng8dl4gps21kfjb0dc3a" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

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

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

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

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

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

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

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

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

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>2erhjxw9ttxng8dl4gps21kfjb0dc3a</sha1>
    </revision>
    <revision>
      <id>2805882</id>
      <parentid>2805881</parentid>
      <timestamp>2026-04-22T06:12:58Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 120-cell */</comment>
      <origin>2805882</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40390" sha1="icn38zp7pqfuv46ggcr4lg4hyw3a9ai" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot;
|[[File:120-cell.gif|200px]]&lt;br&gt;Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
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|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
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|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>icn38zp7pqfuv46ggcr4lg4hyw3a9ai</sha1>
    </revision>
    <revision>
      <id>2805883</id>
      <parentid>2805882</parentid>
      <timestamp>2026-04-22T06:14:21Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40402" sha1="kkihjcek77uirl4s1x0ou7zhwh582g1" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|[[File:120-cell.gif|200px]]&lt;br&gt;Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>kkihjcek77uirl4s1x0ou7zhwh582g1</sha1>
    </revision>
    <revision>
      <id>2805885</id>
      <parentid>2805883</parentid>
      <timestamp>2026-04-22T06:15:53Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40414" sha1="0giqcynbqugy22voabbjykk37i7xnsy" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot; align=&quot;top&quot;
|[[File:120-cell.gif|200px]]&lt;br&gt;Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>0giqcynbqugy22voabbjykk37i7xnsy</sha1>
    </revision>
    <revision>
      <id>2805887</id>
      <parentid>2805885</parentid>
      <timestamp>2026-04-22T06:20:44Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40429" sha1="3erw7amtb0ct3lh58nm95d4s1gnjkdy" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot; style=&quot;vertical-align:top&quot;
|[[File:120-cell.gif|200px]]&lt;br&gt;Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>3erw7amtb0ct3lh58nm95d4s1gnjkdy</sha1>
    </revision>
    <revision>
      <id>2805888</id>
      <parentid>2805887</parentid>
      <timestamp>2026-04-22T06:21:23Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40456" sha1="8b3hfoibnze7flwm0xe0r1e51bxbj5d" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthogonal projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>8b3hfoibnze7flwm0xe0r1e51bxbj5d</sha1>
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    <revision>
      <id>2805890</id>
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      <timestamp>2026-04-22T06:29:53Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40556" sha1="ccb8428iqi1ah0icmvl6gj8gpezskp2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ccb8428iqi1ah0icmvl6gj8gpezskp2</sha1>
    </revision>
    <revision>
      <id>2805893</id>
      <parentid>2805890</parentid>
      <timestamp>2026-04-22T06:42:49Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 120-cell */</comment>
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      <text bytes="40506" sha1="po43cy3w0q5cqh067st35y5pr7qnhc6" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of irregular polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>po43cy3w0q5cqh067st35y5pr7qnhc6</sha1>
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    <revision>
      <id>2805901</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="40496" sha1="cwu7vtfnri1z0ndd6jhc140013i4l43" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except for regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the other 5 regular convex 4-polytopes]].

{{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}}

=== How many building blocks, how many ways ===

The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>cwu7vtfnri1z0ndd6jhc140013i4l43</sha1>
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    <revision>
      <id>2805904</id>
      <parentid>2805901</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Compounds in the 120-cell */</comment>
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      <text bytes="39382" sha1="igt9s2qvqdjb0vvqmasb12ae1u84l28" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's distinct star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their regular star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== The 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The rendering to the left shows only the 120-cell's shortest chords, its 1200 edges. This rendering shows only the 120-cell's longest chords, the 1200 edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Some are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.

Most renderings of the 120-cell, like the rotating animation here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but a total of [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at each vertex.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 discrete chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. Consequently these chords may be considered the 30 most significant discrete distances in geometry.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

...

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. 

[[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|Orthogonal projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;,{{Sfn|Ruen: Great grand stellated 120-cell|2007}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]]
The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) distinct instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]], the final regular [[W:Stellation|stellation]] of the 120-cell, the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside.

The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this planar octagon in 3-space, we can reposition its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can reposition its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs.

The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate, but the most circular kind of rotation (as opposed to an elliptical double rotation) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called isoclinic.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because each of the 16-cell's 6 great square planes has just one other completely orthogonal great square plane. In the 16-cell an isoclinic rotation by 90° in any pair of completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to the position 180° degrees away.

...

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&gt;, so the unit-edge [[w:Tesseract|4-cube (the 8-cell tesseract)]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge.

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star ''n''-polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>igt9s2qvqdjb0vvqmasb12ae1u84l28</sha1>
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  </page>
  <page>
    <title>Patriarch Ages Curious Numerical Facts Response</title>
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    <id>328204</id>
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      <id>2805809</id>
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      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Similarities */</comment>
      <origin>2805809</origin>
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      <text bytes="99757" sha1="0k0wkqwppj7yjcsd2ovg5iv49xfs1ya" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than those of the adjacent patriarchs.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805810</id>
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      <timestamp>2026-04-21T18:53:39Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
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      <comment>/* Masoretic Adjustments */</comment>
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      <text bytes="99865" sha1="ica53vk9v1xlh41d878pjkfmlla7dhx" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than those of the adjacent patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch; whos respective fathering ages are 130, 105, 90, 70, 65, 65.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805811</id>
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      <timestamp>2026-04-21T18:55:30Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
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      <comment>/* Masoretic Adjustments */</comment>
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      <text bytes="99936" sha1="179soic2227s4jok6uasxosu4umvxxc" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than those of the adjacent patriarchs '''Adam''', '''Seth''', '''Enosh''', '''Kenan''', '''Mahalalel''', and ''Enoch'''; whos respective fathering ages are '''130''', '''105''', '''90''', '''70''', '''65''', '''65'''.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    <revision>
      <id>2805813</id>
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      <timestamp>2026-04-21T18:58:41Z</timestamp>
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      <comment>/* Masoretic Adjustments */</comment>
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      <text bytes="99893" sha1="ofmenzm705fhohcl8tk9n4ynxuvq0qa" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below).

In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than those of the adjacent patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch; who were respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65'''.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805814</id>
      <parentid>2805813</parentid>
      <timestamp>2026-04-21T18:58:57Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
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      <comment>/* Masoretic Adjustments */</comment>
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      <text bytes="99892" sha1="lxyww82mjbfgo4yyrb1cku0nmyndf7a" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than those of the adjacent patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch; who were respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65'''.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>lxyww82mjbfgo4yyrb1cku0nmyndf7a</sha1>
    </revision>
    <revision>
      <id>2805815</id>
      <parentid>2805814</parentid>
      <timestamp>2026-04-21T19:00:35Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Masoretic Adjustments */</comment>
      <origin>2805815</origin>
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      <format>text/x-wiki</format>
      <text bytes="99879" sha1="at14ktucnq1z6kb8vo5auusowmiubwh" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than adjacent patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch; who were respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65'''.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    <revision>
      <id>2805816</id>
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      <text bytes="99878" sha1="j1dgf6812jz1fizchmyi7gxzgcy222o" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than adjacent patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch; who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65'''.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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      <id>2805817</id>
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      <text bytes="99905" sha1="16x5oybb0cghamzljuinn7dtdfyptp8" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805818</id>
      <parentid>2805817</parentid>
      <timestamp>2026-04-21T19:10:01Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
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      <comment>/* Flood Adjustment Summary */</comment>
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      <text bytes="99968" sha1="g3fsqq9pa3vpnirc8o33flv2tyzbsjw" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in predictable patterns that reveal the editorial intent of later redactors, as shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805819</id>
      <parentid>2805818</parentid>
      <timestamp>2026-04-21T19:13:28Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
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      <comment>/* Additional Textual Evidence */</comment>
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      <text bytes="99991" sha1="kb6c5vg8dkbg43wgz9wlxawf7uslpqm" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>kb6c5vg8dkbg43wgz9wlxawf7uslpqm</sha1>
    </revision>
    <revision>
      <id>2805847</id>
      <parentid>2805819</parentid>
      <timestamp>2026-04-22T01:47:54Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <format>text/x-wiki</format>
      <text bytes="100201" sha1="id2jhuxv3op8fh9iwimh6oe7jffpzll" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[supplementary hypothesis]], a popular model of the [[composition of the Torah]]. The Priestly source is shown as '''P'''.]]
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>id2jhuxv3op8fh9iwimh6oe7jffpzll</sha1>
    </revision>
    <revision>
      <id>2805848</id>
      <parentid>2805847</parentid>
      <timestamp>2026-04-22T01:49:13Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
      <origin>2805848</origin>
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      <format>text/x-wiki</format>
      <text bytes="100255" sha1="i81uy2ylmnqcmqt72azyml6oy5et9ap" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the Pentateuch (Torah) reached its final form during or shortly after the exile. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>i81uy2ylmnqcmqt72azyml6oy5et9ap</sha1>
    </revision>
    <revision>
      <id>2805849</id>
      <parentid>2805848</parentid>
      <timestamp>2026-04-22T01:55:06Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <text bytes="100303" sha1="jk77mj42svawp6ii6khrdnclh0t754i" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the biblical chronology was developed by an individual or school of scribes know as the [[w:Priestly_source|&quot;Priestly source&quot;]]. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>jk77mj42svawp6ii6khrdnclh0t754i</sha1>
    </revision>
    <revision>
      <id>2805850</id>
      <parentid>2805849</parentid>
      <timestamp>2026-04-22T01:58:11Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <text bytes="100326" sha1="nam6yjp35ob7a1ijrh6vo6uajppbcmq" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the biblical chronology was developed and added to the text by an individual or school of scribes known as the [[w:Priestly_source|&quot;Priestly source&quot;]]. Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>nam6yjp35ob7a1ijrh6vo6uajppbcmq</sha1>
    </revision>
    <revision>
      <id>2805851</id>
      <parentid>2805850</parentid>
      <timestamp>2026-04-22T01:59:54Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
      <origin>2805851</origin>
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      <text bytes="100353" sha1="8kajy2x2hbkne3u1ikdqukb9zp7z64e" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]
The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). As the captivity transitioned Israelite religion from a localized temple cult into a 'portable' Judaism centered on sacred text, Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

Most scholars believe the biblical chronology was developed and added to the text by an individual or school of scribes known as the [[w:Priestly_source|&quot;Priestly source&quot;]] (see diagram to the right). Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. This process involved not just preservation, but an intellectual synchronization with Mesopotamian standards of antiquity. The following comparisons illustrate how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology.

The following bullet point list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>8kajy2x2hbkne3u1ikdqukb9zp7z64e</sha1>
    </revision>
    <revision>
      <id>2805852</id>
      <parentid>2805851</parentid>
      <timestamp>2026-04-22T02:05:59Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <format>text/x-wiki</format>
      <text bytes="100095" sha1="i01ez9ytautq7vk6ibhymxl5pgpqkdt" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Most scholars believe the biblical chronology was developed and added to the text by an individual or school of scribes known as the [[w:Priestly_source|&quot;Priestly source&quot;]] (see diagram to the right). Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

The use of Mesopotamian (sexagesimal) numbers in the Prototype 1 Chronology provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). The following bullet point list illustrates how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology. The list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration groupings to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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      <text bytes="100065" sha1="h9m5ric5pdnth26wvsq68of2n26wxbv" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Most scholars believe the biblical chronology was developed and added to the text by an individual or school of scribes known as the [[w:Priestly_source|&quot;Priestly source&quot;]] (see diagram to the right). Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context.

The use of Mesopotamian (sexagesimal) numbers provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE). The following bullet point list illustrates how Prototype 1 lifespans utilized the mathematical logic of the Sumerian King List to establish a prestigious, foundational chronology. The list is organized by time duration, beginning with the longest lifespans found in the '''Prototype 1''' Chronology. Lifespan length groupings are paired with kingship duration groupings to show comparable figures.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>h9m5ric5pdnth26wvsq68of2n26wxbv</sha1>
    </revision>
    <revision>
      <id>2805854</id>
      <parentid>2805853</parentid>
      <timestamp>2026-04-22T02:24:42Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <text bytes="99952" sha1="jqt8dggz3s6iydgzxdtmxfzztfuwrvg" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Most scholars believe the biblical chronology was developed and added to the text by an individual or school of scribes known as the [[w:Priestly_source|&quot;Priestly source&quot;]] (see diagram to the right). Deprived of a physical Temple for sacrifice, the Judean elite focused on transforming oral and fragmented traditions into a permanent, 'portable' written Law. Judean scribes likely adopted the prestigious base-60 mathematical system of their captors to codify a history that commanded respect in a Mesopotamian intellectual context. The use of Mesopotamian (sexagesimal) numbers provides strong evidence that these lifespans were integrated into biblical narratives during the Babylonian captivity (c. 586–538 BCE).

The following bullet points illustrate how the Prototype 1 chronology utilized timespans that are found in the Sumerian King List. The longest lifespans in the '''Prototype 1''' Chronology, 960 years and 900 years, are figures which are well represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]
*** [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs)
*** Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]
*** [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch)
*** Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]
*** [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs)
*** Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically structured to mirror them.&quot;

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>jqt8dggz3s6iydgzxdtmxfzztfuwrvg</sha1>
    </revision>
    <revision>
      <id>2805855</id>
      <parentid>2805854</parentid>
      <timestamp>2026-04-22T02:34:18Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <text bytes="99912" sha1="9v7j74nayzbted13o36d0jps5rovahf" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs): Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs): Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]: [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch): Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs): Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>9v7j74nayzbted13o36d0jps5rovahf</sha1>
    </revision>
    <revision>
      <id>2805857</id>
      <parentid>2805855</parentid>
      <timestamp>2026-04-22T02:42:40Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
      <origin>2805857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="100186" sha1="48gtfgmgwk1i36kbg1xi9yde00jmpyy" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''' Chronology (4 Patriarchs): Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs): Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]: [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch): Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs): Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>48gtfgmgwk1i36kbg1xi9yde00jmpyy</sha1>
    </revision>
    <revision>
      <id>2805858</id>
      <parentid>2805857</parentid>
      <timestamp>2026-04-22T02:44:44Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
      <origin>2805858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="100075" sha1="8wtagekjrd95td8hmmbsom8i17casm0" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Four Kings)]]: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''' Chronology (4 Patriarchs): Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (One King)]]: [[w:Atab|Atab]]
** '''Prototype 1''' Chronology (1 Patriarch): Shem

* '''7 ''šūši'' (420 years)'''
** [[w:Sumerian_King_List#Rulers_in_the_Sumerian_King_List|Sumerian King List (Two Kings)]]: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''' Chronology (2 Patriarchs): Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805859</id>
      <parentid>2805858</parentid>
      <timestamp>2026-04-22T02:46:16Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Mesopotamian Derived Lifespans */</comment>
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      <text bytes="99743" sha1="s3jasncgickvorklwj2gu8lp7zar0hn" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], Adapa is linked to the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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      <id>2805860</id>
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      <text bytes="99789" sha1="e3c7r16ztliezyzcih1ff3p2ptmy6ee" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    <revision>
      <id>2805861</id>
      <parentid>2805860</parentid>
      <timestamp>2026-04-22T03:02:15Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
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      <comment>/* Septuagint Adjustments */</comment>
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      <text bytes="99799" sha1="6kqrprdll0vd3zz1z220vy5uogtcaxc" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2805862</id>
      <parentid>2805861</parentid>
      <timestamp>2026-04-22T03:08:55Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Lifespan Adjustments by Individual Patriarch */</comment>
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      <text bytes="99799" sha1="8q9aznt6xgcsz1vks9svanx4kb10d36" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
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      <id>2805863</id>
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        <username>CanonicalMormon</username>
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      <comment>/* Additional Textual Evidence */</comment>
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      <text bytes="99799" sha1="tlxop17pek9zmmw8klg8uzrl8s8syv7" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As shown in the above table, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, as noted previously, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    <revision>
      <id>2805864</id>
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      <timestamp>2026-04-22T03:14:17Z</timestamp>
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      <text bytes="99770" sha1="ryiwe7bse3zzx1t5ogvagc1l481agoa" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;4&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;5&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| colspan = &quot;3&quot; | Varied &lt;br/&gt; &lt;small&gt;(1454 / 1474)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(Varied)&lt;br/&gt;(Varied)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;3&quot; |&lt;small&gt;(Varied)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;3&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter.”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962
| 847
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969
| 720
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| 777
| 653
| 707
| 723
| 753

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600
| rowspan=&quot;9&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438
| 538
| 535

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| —
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433
| 536
| 460

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464
| 404
| 567
| colspan=&quot;2&quot; | 404

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| colspan=&quot;2&quot; | 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239
| 342
| 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230
| colspan=&quot;2&quot; | 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148
| 198
| 304

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205
| 145
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180
| 185
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137
| rowspan=&quot;3&quot; | —
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131
| 137
| 136
| colspan=&quot;2&quot; | 132

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| —
| colspan=&quot;1&quot; | 13,200
| colspan=&quot;1&quot; | 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Josephus&lt;br/&gt;(94 AD)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2626&lt;br/&gt;&lt;small&gt;(2702 - 76)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
|-

| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | —
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| —
| 13,200
| 13,551
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

While the chronologies of the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' are anchored in Levant-based agricultural cycles and the symbolic interplay of the numbers 40 and 49, the Septuagint (LXX) appears to have been structured around a different set of priorities. Specifically, the LXX's chronological framework seems designed to resolve a significant textual difficulty: the mathematical anomaly of patriarchs potentially outliving the Flood. In the 2017 article, ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', author Paul D. makes the following statement regarding the Septuagint:

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

While Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor intended to &quot;fix&quot; the timeline but failed in the case of Methuselah, this interpretation potentially overlooks the systemic nature of the changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single &quot;failure&quot; to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective.

Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

=== Demetrius the Chronographer ===
Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the &quot;Long Chronology&quot; of the Septuagint.

In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his &quot;Whoops Theory&quot; by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: 

* '''Josephus:''' Characterized as dependent on the Masoretic tradition.
* '''Pseudo-Philo:''' Dismissed due to textual corruption (&quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown.

The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses.

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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      <comment>/* a thing I may regret */ linking to my daily notes</comment>
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      <text bytes="6179" sha1="r90meaor9qu40rhaqx7w7iagxanar0j" xml:space="preserve">== April 20th experiment, &quot;AI Decisions, sure. AI-generation NEVER&quot; ==

Starting today on April 20th after 08:46 UTC Time(I got UTC time on this computer where I'm so far only using this account), I'll begin by editing Wikiversity resources by being more encouraged by &quot;yeah, do that&quot; comments by Large Language Models.

Nothing of it will be &quot;AI-generated&quot; but the decisions I take: the reason for the decisions I take may be because of &quot;AI-generation&quot; but of course I will try to stay away from clear stupidity like if the AI-generation says &quot;jump off a cliff&quot;. An extreme example, but I wanted to make a point that I won't take any decision and I will question the &quot;AI/LLM&quot; if it suggests something that to me sounds insane.

If you see anything weird please comment on my talk page after you've reverted my edits.

When this experiment ends, I don't have a plan for that yet. User input might help.

This is where I make notes of decisions that may motivation me to do edits in places. It should include both inputs and outputs and what kind of &quot;version&quot; of &quot;AI&quot;s/LLMs I'm using:
* [[/April 20th Experiment Notes|&quot;AI Notes&quot; for motivation purposes in this &quot;experiment&quot;]]

== Main focus: my &quot;idea&quot; ==
* This is my [[Draft:The Neurodiversity-inspired Idea]]. There goes the &quot;main effort&quot; based on my other smaller effort in various places and also by using the methodology I one day hope I will make.
* [[User:ThinkingScience/ND_Inspired_Idea_Notebook|Daily Diary of ND Inspired Idea]]

== Coursera schedule and notes ==

Today April 16, 2026 my contributions contain a lot of spelling mistakes. They may be present other days too. You'll probably spot spelling mistakes all over.

My studying schedule as I've understood it so far(studying with my mother):

This schedule is not reliable(cause my studying partner keeps changing the time, which is not necessarily bad):

UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week)
* Monday
* Thursday
* Saturday

These are my course notes: [[User:ThinkingScience/Draftspace/Coursera]]

== I'm studying on Coursera and about their Terms of Use ==
'''Nothing here is legal advice'''. This is very important.
Nothing in this &quot;Wikisection&quot; constitutes legal advice! Please don't blindly follow my advice and if someone copies some parts of this text without providing context then they are responsible for what they share! If you have been tricked by scammers that's sad but I am NOT responsible for illegal activities.

* web.archive.org/web/20260325233813/https://www.coursera.org/about/terms

&quot;When you create your Coursera account, and when you subsequently use certain features, you must provide us with accurate and complete information, and you agree to update your information to keep it accurate and complete.&quot;

My interpretation of that is that on Coursera I have to provide a real name. There is a field for &quot;Full name&quot;(retrieved 2026-04-09 UTC YYYY-MM-DD). How does that correspond to these terms? It doesn't say &quot;Real name&quot; but even if it did, what if I choose a name for myself and I'd like to call myself ThinkingScience? Is it still accurate? 

They don't specify what I actually have to do, just based on my quote. It would be nice for me and other Coursera learners to know what is true. Is the privacy on Wikiversity better? I'd say it is because on Coursera we are forced to provide an email address to create an account. We are not forced to do that on Wikiversity, Wikidata etc.

== notes about this account ==

This account is an alternative account on a computer I don't trust. It should never be allowed to vote and if it does please block this account.

It's an alt of [[User:Dekatriofovia]] which unfortunately I have to prove right now despite me being in a hurry...so I'll edit my account at Dekatriofovia at the same time almost and publish at the same time...so you know it's me.

The reason for this account is it's on a computer with a bigger screen so I can more easily read books and documents.

== a thing I did not regret(modified section title) ==

This may be blathering but it ends with another Wikilink where I will pass my &quot;idea&quot; through '''Wikiversity:Research ethics''' and through anything else that might be required before anything enters Draft space. The &quot;idea&quot; is &quot;'''The Neurodiversity-inspired idea'''&quot;.

[[Protoscience]] was an interesting read. I think it will be calming for me if my idea is proven to be pseudoscience cause I can stop worrying about it and leave it behind me. &quot;The Neurodiversity-inspired idea&quot;(in lack for a better name, for now) will not be published in main space, only in draft space.

[[Wikiversity:Original research]] made me think &quot;I may be way over my head&quot; (though I stumbled around a bit due to not knowing English at an advanced enough level...this parenthesis is about some unimportant trivia).

I'm gonna place everything regarding &quot;The Neurodiversity-inspired idea&quot; into draft space and pass it through '''Wikiversity:Research ethics'''(sorry for repeating myself) and anything else I can find and also ask the community here on Wikiversity what else to place it through.

I thought I was gonna create '''User:ThinkingScience/The Neurodiversity-inspired Idea'''(but turns out I was encouraged to create it in Draft: space ... (this paragraph has been modified. Edit history might keep the original). Here are my notes again which I wanted to link to [[User:ThinkingScience/ND Inspired Idea Notebook]]

'''Draft:The Neurodiversity-inspired Idea''' that probably is in line with &quot;be bold&quot;.

=== It happened, a small burden has been lifted ===

I posted to the [[Wikiversity:Colloquium]] https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;oldid=2805080

Thing may be archive in the future. I've lost many things that way.(but also re-discovered many things that landed in the archive that I had posted too!)

One week. One small burden lifted. It was the only way forward. I may have been driven insane otherwise or this is just a very bad day I'm having. Full of things that &quot;real life&quot; is demanding of me.

More specifically, this is what I posted [[Wikiversity:Colloquium#Advice_needed:_A_Neurodiversity-inspired_Idea/observation]]</text>
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    <revision>
      <id>2805920</id>
      <parentid>2805913</parentid>
      <timestamp>2026-04-22T07:50:19Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>the risk of doing anything in life</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7410" sha1="pimdpnt26oh6a2fspd7rq6u4pg2i9ju" xml:space="preserve">== Taking responsibility for famous people or people to focus on in [[Draft:The Neurodiversity-inspired Idea]] ==

I need to take responsibility for the choices I make. If any of my choices resulted in a harm to a real person I am responsible whether I agree or not to any blame being put on me.

This section may be moved to a sub-page if I think it starts getting too &quot;cluttered&quot; and later into more sub-pages if the list just grows and grows.

=== T ===

Taylor Swift. Why I chose Taylor Swift. I have watched interviews with her before. She is an interesting person to me. I discovered she is open about her creation process. I value that in human beings and that includes people I meet offline, in the &quot;real world&quot; but it will be a challenge for me to make video notes that are &quot;Do no harm&quot;. I may be &quot;way in over my head&quot;. Please help me if you think I'm doing something wrong. 

She has a dedicated follower base which may have a large influence. Maybe I'll suffer for this but keeping my &quot;idea&quot; locked inside &quot;my head&quot; I think will cause greater harm to me than good. I know what risk I am taking...or probably not but I gotta move forward or try to. Perhaps my fears are greater than real risks in reality but who knows?

== April 20th experiment, &quot;AI Decisions, sure. AI-generation NEVER&quot; ==

Starting today on April 20th after 08:46 UTC Time(I got UTC time on this computer where I'm so far only using this account), I'll begin by editing Wikiversity resources by being more encouraged by &quot;yeah, do that&quot; comments by Large Language Models.

Nothing of it will be &quot;AI-generated&quot; but the decisions I take: the reason for the decisions I take may be because of &quot;AI-generation&quot; but of course I will try to stay away from clear stupidity like if the AI-generation says &quot;jump off a cliff&quot;. An extreme example, but I wanted to make a point that I won't take any decision and I will question the &quot;AI/LLM&quot; if it suggests something that to me sounds insane.

If you see anything weird please comment on my talk page after you've reverted my edits.

When this experiment ends, I don't have a plan for that yet. User input might help.

This is where I make notes of decisions that may motivation me to do edits in places. It should include both inputs and outputs and what kind of &quot;version&quot; of &quot;AI&quot;s/LLMs I'm using:
* [[/April 20th Experiment Notes|&quot;AI Notes&quot; for motivation purposes in this &quot;experiment&quot;]]

== Main focus: my &quot;idea&quot; ==
* This is my [[Draft:The Neurodiversity-inspired Idea]]. There goes the &quot;main effort&quot; based on my other smaller effort in various places and also by using the methodology I one day hope I will make.
* [[User:ThinkingScience/ND_Inspired_Idea_Notebook|Daily Diary of ND Inspired Idea]]

== Coursera schedule and notes ==

Today April 16, 2026 my contributions contain a lot of spelling mistakes. They may be present other days too. You'll probably spot spelling mistakes all over.

My studying schedule as I've understood it so far(studying with my mother):

This schedule is not reliable(cause my studying partner keeps changing the time, which is not necessarily bad):

UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week)
* Monday
* Thursday
* Saturday

These are my course notes: [[User:ThinkingScience/Draftspace/Coursera]]

== I'm studying on Coursera and about their Terms of Use ==
'''Nothing here is legal advice'''. This is very important.
Nothing in this &quot;Wikisection&quot; constitutes legal advice! Please don't blindly follow my advice and if someone copies some parts of this text without providing context then they are responsible for what they share! If you have been tricked by scammers that's sad but I am NOT responsible for illegal activities.

* web.archive.org/web/20260325233813/https://www.coursera.org/about/terms

&quot;When you create your Coursera account, and when you subsequently use certain features, you must provide us with accurate and complete information, and you agree to update your information to keep it accurate and complete.&quot;

My interpretation of that is that on Coursera I have to provide a real name. There is a field for &quot;Full name&quot;(retrieved 2026-04-09 UTC YYYY-MM-DD). How does that correspond to these terms? It doesn't say &quot;Real name&quot; but even if it did, what if I choose a name for myself and I'd like to call myself ThinkingScience? Is it still accurate? 

They don't specify what I actually have to do, just based on my quote. It would be nice for me and other Coursera learners to know what is true. Is the privacy on Wikiversity better? I'd say it is because on Coursera we are forced to provide an email address to create an account. We are not forced to do that on Wikiversity, Wikidata etc.

== notes about this account ==

This account is an alternative account on a computer I don't trust. It should never be allowed to vote and if it does please block this account.

It's an alt of [[User:Dekatriofovia]] which unfortunately I have to prove right now despite me being in a hurry...so I'll edit my account at Dekatriofovia at the same time almost and publish at the same time...so you know it's me.

The reason for this account is it's on a computer with a bigger screen so I can more easily read books and documents.

== a thing I did not regret(modified section title) ==

This may be blathering but it ends with another Wikilink where I will pass my &quot;idea&quot; through '''Wikiversity:Research ethics''' and through anything else that might be required before anything enters Draft space. The &quot;idea&quot; is &quot;'''The Neurodiversity-inspired idea'''&quot;.

[[Protoscience]] was an interesting read. I think it will be calming for me if my idea is proven to be pseudoscience cause I can stop worrying about it and leave it behind me. &quot;The Neurodiversity-inspired idea&quot;(in lack for a better name, for now) will not be published in main space, only in draft space.

[[Wikiversity:Original research]] made me think &quot;I may be way over my head&quot; (though I stumbled around a bit due to not knowing English at an advanced enough level...this parenthesis is about some unimportant trivia).

I'm gonna place everything regarding &quot;The Neurodiversity-inspired idea&quot; into draft space and pass it through '''Wikiversity:Research ethics'''(sorry for repeating myself) and anything else I can find and also ask the community here on Wikiversity what else to place it through.

I thought I was gonna create '''User:ThinkingScience/The Neurodiversity-inspired Idea'''(but turns out I was encouraged to create it in Draft: space ... (this paragraph has been modified. Edit history might keep the original). Here are my notes again which I wanted to link to [[User:ThinkingScience/ND Inspired Idea Notebook]]

'''Draft:The Neurodiversity-inspired Idea''' that probably is in line with &quot;be bold&quot;.

=== It happened, a small burden has been lifted ===

I posted to the [[Wikiversity:Colloquium]] https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&amp;oldid=2805080

Thing may be archive in the future. I've lost many things that way.(but also re-discovered many things that landed in the archive that I had posted too!)

One week. One small burden lifted. It was the only way forward. I may have been driven insane otherwise or this is just a very bad day I'm having. Full of things that &quot;real life&quot; is demanding of me.

More specifically, this is what I posted [[Wikiversity:Colloquium#Advice_needed:_A_Neurodiversity-inspired_Idea/observation]]</text>
      <sha1>pimdpnt26oh6a2fspd7rq6u4pg2i9ju</sha1>
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  </page>
  <page>
    <title>Draft:The Neurodiversity-inspired Idea</title>
    <ns>118</ns>
    <id>329157</id>
    <revision>
      <id>2805907</id>
      <parentid>2805741</parentid>
      <timestamp>2026-04-22T07:19:48Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Questions that might encourage the development of this idea and its methodology */ I am taking to heart some suggestions at the Colloquium. Making example &quot;name ideas&quot;</comment>
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      <text bytes="4661" sha1="p0gsboea66m2ogm9tajiq1i40n3x192" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access to the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===
* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Psychiatric Spectrum Specifications
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>p0gsboea66m2ogm9tajiq1i40n3x192</sha1>
    </revision>
    <revision>
      <id>2805909</id>
      <parentid>2805907</parentid>
      <timestamp>2026-04-22T07:21:36Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Naming Suggestions */ may make even less sense but I felt I needed to add that. I feel this is a &quot;pre-research&quot; thing. Figuring out if this can be made into hypotheses</comment>
      <origin>2805909</origin>
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      <format>text/x-wiki</format>
      <text bytes="4723" sha1="3lrhk52dt0f9ie7cgrggzticsy92dmi" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access to the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===
* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>3lrhk52dt0f9ie7cgrggzticsy92dmi</sha1>
    </revision>
    <revision>
      <id>2805923</id>
      <parentid>2805909</parentid>
      <timestamp>2026-04-22T08:08:59Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;Research projects must fully document the methods&quot; */ 29 years of age, minimum age for participation in this project? Bad idea? Feedback?</comment>
      <origin>2805923</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5283" sha1="5snl1vwd6kq46ovz4g4gb40qkijwvbg" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

An idea for the minimum age of the subjects which are &quot;studied&quot; in a &quot;Do no harm&quot; way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the &quot;starting age&quot; or &quot;minimum age&quot; is high enough so that a person with a &quot;fully matured brain&quot; also has some experience living with that fully matured brain.

💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of &quot;Do no harm&quot;?

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access to the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===
* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>5snl1vwd6kq46ovz4g4gb40qkijwvbg</sha1>
    </revision>
    <revision>
      <id>2805924</id>
      <parentid>2805923</parentid>
      <timestamp>2026-04-22T08:10:19Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;Research projects must fully document the methods&quot; */ consent</comment>
      <origin>2805924</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5330" sha1="j9reodzi42ewf6icygy4r0og2twfi9a" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

An idea for the minimum age of the subjects which are &quot;studied&quot; in a &quot;Do no harm&quot; way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the &quot;starting age&quot; or &quot;minimum age&quot; is high enough so that a person with a &quot;fully matured brain&quot; also has some experience living with that fully matured brain.

💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of &quot;Do no harm&quot;?

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access and specific consent from any person appearing in the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===
* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>j9reodzi42ewf6icygy4r0og2twfi9a</sha1>
    </revision>
  </page>
  <page>
    <title>User:Atcovi/OGM &amp; Suicide</title>
    <ns>2</ns>
    <id>329164</id>
    <revision>
      <id>2805765</id>
      <parentid>2805716</parentid>
      <timestamp>2026-04-21T13:39:34Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Moore &amp; Zoellner (2007) – Evaluative Review (OGM &amp; trauma) */</comment>
      <origin>2805765</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9512" sha1="kccg01qbk4ut6h2se8rg859bmt5u8oy" xml:space="preserve">== Research ==

=== [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] ===
'''Background'''

* Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner&lt;ref&gt;The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.&lt;/ref&gt;.
* Mentions number of studies associating OGM with suicide: &quot;''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''&quot;
* [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓
* '''Research question(s)?''' No clear evidence to show how childhood trauma &amp; OGM interact in the suicidal process and whether depression is a ''moderating'' effect. 
* '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals.
* '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways.

'''Method'''

* '''356 Chinese participants'''.
** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. 
** '''176 healthy individuals'''
'''Measures'''

* '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. 
* '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. 
* '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. 
* '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38.  
** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. 
** ↑ BSI-CV score ↑ SI
* '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: &quot;how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?&quot;.
'''Results'''

* Trauma → OGM, WSI, CSI
* OGM → WSI + CSI
* WSI → CSI
* WSI → PSB

* PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]'''

WSI is strongly predictive of:

* PSB
* future SI intensity 

'''Discussion'''

# '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically.
# '''WSI''' = worst suicidal thoughts one has ever had [at a certain point].
# '''CSI''' = current point of suicidal ideation
* [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together.
* '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. 
** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI.
** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. &lt;-- appears to be relevant only with other paired vulnerabilities (ex, trauma).
** OGM as a stable trait of depression was more severe in the depression group vs. healthy control.
** Childhood trauma &amp; OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model.
** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity.
** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation.
** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) &amp; WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation.
* Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy &amp; MeST, both geared towards recalling memories in detail.
* '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors.
* '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior.

=== [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] ===
Crane et al. (2016) – Community adolescent cohort

* Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16)
* Tested whether OGM predicts depression/suicidality and moderates life stress
* '''Findings:'''
** OGM did not predict depression, suicidal ideation, or self-harm
** OGM did not moderate the effect of life events
** Life events strongly predicted all outcomes
* '''Interpretation:'''
** OGM may not function as a general vulnerability factor in community samples
** Likely only relevant in high-risk or depressed populations

=== Sumner et al. (2010) – Meta-analysis (OGM &amp; depression) ===

* Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression
* '''Findings:'''
** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up
** Effect remains '''even after controlling for baseline depression'''
** Overall effect size is '''small but significant (~1–2% variance explained)'''
* '''Moderators:'''
** Stronger effect in '''clinically depressed samples vs nonclinical'''
** Stronger with '''shorter follow-up periods'''

* '''Interpretation:'''
** OGM is a '''predictor of depression maintenance''', not just a correlate
** Likely acts as a '''vulnerability factor''', especially in high-risk groups
{{Notice|OGM works under certain conditions (clinical / high-risk), not universally}}

=== Moore &amp; Zoellner (2007) – Evaluative Review (OGM &amp; trauma) ===

* Evaluative review of '''24 studies''' examining trauma exposure and OGM
* '''Core question:''' Does trauma cause overgeneral memory?
'''Findings'''
* ❌ '''No consistent link''' between trauma exposure and OGM
* ✅ OGM is more consistently linked to:
** '''Depression'''
** '''PTSD symptoms (intrusions, avoidance)'''
* Trauma exposure alone is '''not sufficient''' to produce OGM
* OGM appears more tied to '''psychopathology''', not the event itself
'''Key nuance'''
* Post-trauma '''symptoms''' (not the trauma itself) are what matter
* Evidence across studies is '''mixed and methodologically inconsistent'''
'''Interpretation'''
* Challenges the classic “trauma → OGM” theory
* Supports alternative view:&lt;blockquote&gt;OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)&lt;/blockquote&gt;
*Trauma alone isn’t enough — psychological response matters

== Narratives? ==

== See also ==

* [[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]]

== Notes ==
{{Reflist}}</text>
      <sha1>kccg01qbk4ut6h2se8rg859bmt5u8oy</sha1>
    </revision>
    <revision>
      <id>2805801</id>
      <parentid>2805765</parentid>
      <timestamp>2026-04-21T17:23:09Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Research */</comment>
      <origin>2805801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15238" sha1="498c3nadoqjjhx4n7i6e5g3v77p5hbh" xml:space="preserve">== Research ==

=== [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] ===
'''Background'''

* Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner&lt;ref&gt;The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.&lt;/ref&gt;.
* Mentions number of studies associating OGM with suicide: &quot;''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''&quot;
* [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓
* '''Research question(s)?''' No clear evidence to show how childhood trauma &amp; OGM interact in the suicidal process and whether depression is a ''moderating'' effect. 
* '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals.
* '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways.

'''Method'''

* '''356 Chinese participants'''.
** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. 
** '''176 healthy individuals'''
'''Measures'''

* '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. 
* '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. 
* '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. 
* '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38.  
** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. 
** ↑ BSI-CV score ↑ SI
* '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: &quot;how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?&quot;.
'''Results'''

* Trauma → OGM, WSI, CSI
* OGM → WSI + CSI
* WSI → CSI
* WSI → PSB

* PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]'''

WSI is strongly predictive of:

* PSB
* future SI intensity 

'''Discussion'''

# '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically.
# '''WSI''' = worst suicidal thoughts one has ever had [at a certain point].
# '''CSI''' = current point of suicidal ideation
* [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together.
* '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. 
** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI.
** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. &lt;-- appears to be relevant only with other paired vulnerabilities (ex, trauma).
** OGM as a stable trait of depression was more severe in the depression group vs. healthy control.
** Childhood trauma &amp; OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model.
** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity.
** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation.
** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) &amp; WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation.
* Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy &amp; MeST, both geared towards recalling memories in detail.
* '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors.
* '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior.

=== [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] ===
Crane et al. (2016) – Community adolescent cohort

* Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16)
* Tested whether OGM predicts depression/suicidality and moderates life stress
* '''Findings:'''
** OGM did not predict depression, suicidal ideation, or self-harm
** OGM did not moderate the effect of life events
** Life events strongly predicted all outcomes
* '''Interpretation:'''
** OGM may not function as a general vulnerability factor in community samples
** Likely only relevant in high-risk or depressed populations

=== Sumner et al. (2010) – Meta-analysis (OGM &amp; depression) ===

* Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression
* '''Findings:'''
** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up
** Effect remains '''even after controlling for baseline depression'''
** Overall effect size is '''small but significant (~1–2% variance explained)'''
* '''Moderators:'''
** Stronger effect in '''clinically depressed samples vs nonclinical'''
** Stronger with '''shorter follow-up periods'''

* '''Interpretation:'''
** OGM is a '''predictor of depression maintenance''', not just a correlate
** Likely acts as a '''vulnerability factor''', especially in high-risk groups
{{Notice|OGM works under certain conditions (clinical / high-risk), not universally}}

=== Moore &amp; Zoellner (2007) – Evaluative Review (OGM &amp; trauma) ===

* Evaluative review of '''24 studies''' examining trauma exposure and OGM
* '''Core question:''' Does trauma cause overgeneral memory?
'''Findings'''
* ❌ '''No consistent link''' between trauma exposure and OGM
* ✅ OGM is more consistently linked to:
** '''Depression'''
** '''PTSD symptoms (intrusions, avoidance)'''
* Trauma exposure alone is '''not sufficient''' to produce OGM
* OGM appears more tied to '''psychopathology''', not the event itself
'''Key nuance'''
* Post-trauma '''symptoms''' (not the trauma itself) are what matter
* Evidence across studies is '''mixed and methodologically inconsistent'''
'''Interpretation'''
* Challenges the classic “trauma → OGM” theory
* Supports alternative view:&lt;blockquote&gt;OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)&lt;/blockquote&gt;
*Trauma alone isn’t enough — psychological response matters

=== Crane &amp; Duggan (2009) – CSA onset &amp; OGM in suicidal patients ===

* Clinical sample of '''49 patients with recurrent suicidal behavior'''
* Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM

'''Findings'''

* Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories)
* Effect remained after controlling for:
** depression
** verbal fluency
* Presence vs absence of CSA '''did NOT differ in OGM'''
** (because sample already highly clinical)

'''Interpretation'''

* Not trauma itself → but '''timing of trauma (early development)''' matters
* Supports idea that OGM develops when:
** memory systems are still forming
** avoidance becomes a learned coping style 

'''Insights'''

# Analysis of OGM in a suicidal population.
# Moore &amp; Zoellner had trauma alone ≠ OGM, while Crane &amp; Duggan added some trauma characteristics from early developmental trauma. 
# ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' 

=== Champagne et al. (2016) – OGM in adolescent MDD ===

* Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed)
* Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)'''

'''Findings:'''

* Adolescents with '''current AND remitted MDD''' showed more OGM than controls
* No significant difference between current vs remitted groups
* OGM present for both '''positive and negative cues'''

'''Interpretation:'''

* OGM is likely a '''trait-like vulnerability''', not just a temporary symptom
* May contribute to '''risk of recurrence''' in depression

'''1. Trait vs state question (very important)'''

This paper directly answers:&lt;blockquote&gt;Is OGM just a symptom or a vulnerability?&lt;/blockquote&gt;👉 Answer: '''likely vulnerability'''

That’s big for your argument.
----'''2. Adolescent + clinical sample'''

* Not just adults
* Not just community

👉 Stronger relevance than many papers
----'''3. Fits your mechanism pathway'''

Supports:&lt;blockquote&gt;OGM persists → increases long-term risk → can feed into suicidal ideation&lt;/blockquote&gt;'''Limitations'''

* Small sample (n=65 total)
* Cross-sectional → can’t prove causality

* Can’t tell if OGM came '''before''' depression or is a “scar”

=== Arie et al. (2008) – OGM, problem solving &amp; suicidal behavior ===

* Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls)
* Tested Williams’ model linking OGM, problem solving, and suicide

'''Findings:'''

* Suicidal adolescents showed:
** '''More OGM (less specific memory)'''
** '''Worse interpersonal problem-solving ability'''
** '''Higher hopelessness'''
* OGM significantly correlated with:
** poorer problem solving
** higher hopelessness
** greater suicidal behavior
* Negative childhood life events also linked to OGM and suicide

'''Interpretation:'''

* Supports pathway:&lt;blockquote&gt;OGM → poor problem solving → hopelessness → suicidal behavior&lt;/blockquote&gt;
* OGM may limit access to specific past experiences needed to solve problems effectively

'''Insights'''

* Study on population actually attempting suicide.
* Points out a clear pathway
* Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness

'''Limitations''' 

* n=75, small sample
* cross sectional, so causality can not be pinpointed
* inpatient sample, so cannot be generalizable

=== Kaviani et al. (2011) – OGM, problem solving &amp; suicidal ideation ===

* Clinical sample: depressed patients '''with vs without suicidal ideation'''
* Tested links between OGM, problem solving, depression, and hopelessness

'''Findings:'''

* Suicidal ideation group:
** '''Less specific autobiographical memories (more OGM)'''
** '''More hopelessness'''
** '''Less effective problem-solving strategies'''
* OGM associated with:
** poorer problem solving
** greater hopelessness
* Evidence for a “cycle”:&lt;blockquote&gt;OGM → poor problem solving → hopelessness → suicidal ideation&lt;/blockquote&gt;

'''Interpretation:'''

* OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving
* Supports a '''cognitive pathway to suicide risk'''

'''Insights'''

* Studies suicidal ideation directly.
* Mechanism displayed: OGM --&gt; problem solving --&gt; hopelessness --&gt; suicidal ideation
* Within-depression comparison

'''Limitations'''

* n=40
* Cross-sectional, lacks causality
* Specific country (Iran)

=== Zhu et al. (2025) – OGM &amp; suicidal ideation (ML study) ===

* Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels)
* Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation

'''Findings:'''

* Patients with suicidal ideation:
** '''More OGM (fewer specific memories)'''
** Especially impaired recall of '''positive memories'''
* OGM strongly associated with '''severity of suicidal ideation'''
* Crucially:
** '''OGM predicts suicidal ideation independently of depression severity'''
* ML model showed:
** OGM features were key for identifying '''presence of suicidal ideation'''
** Depression severity features ≠ sufficient to predict SI

'''Interpretation:'''

* OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression
* Supports:&lt;blockquote&gt;OGM contributes uniquely to suicidal ideation beyond general depression&lt;/blockquote&gt;'''Insights'''

* OGM is unique in contribution/specifically tied to SI.
* Separated depression anxiety and suicidal ideation.

OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation.

== Narratives? ==

== See also ==

* [[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]]

== Notes ==
{{Reflist}}</text>
      <sha1>498c3nadoqjjhx4n7i6e5g3v77p5hbh</sha1>
    </revision>
    <revision>
      <id>2805802</id>
      <parentid>2805801</parentid>
      <timestamp>2026-04-21T17:23:39Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>paper count</comment>
      <origin>2805802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15270" sha1="63wpey2abpq94y29diohohz4msc25el" xml:space="preserve">''Paper count:'' 9 (4/21/2026)

== Research ==

=== [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] ===
'''Background'''

* Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner&lt;ref&gt;The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.&lt;/ref&gt;.
* Mentions number of studies associating OGM with suicide: &quot;''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''&quot;
* [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓
* '''Research question(s)?''' No clear evidence to show how childhood trauma &amp; OGM interact in the suicidal process and whether depression is a ''moderating'' effect. 
* '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals.
* '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways.

'''Method'''

* '''356 Chinese participants'''.
** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. 
** '''176 healthy individuals'''
'''Measures'''

* '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. 
* '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. 
* '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. 
* '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38.  
** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. 
** ↑ BSI-CV score ↑ SI
* '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: &quot;how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?&quot;.
'''Results'''

* Trauma → OGM, WSI, CSI
* OGM → WSI + CSI
* WSI → CSI
* WSI → PSB

* PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]'''

WSI is strongly predictive of:

* PSB
* future SI intensity 

'''Discussion'''

# '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically.
# '''WSI''' = worst suicidal thoughts one has ever had [at a certain point].
# '''CSI''' = current point of suicidal ideation
* [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together.
* '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. 
** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI.
** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. &lt;-- appears to be relevant only with other paired vulnerabilities (ex, trauma).
** OGM as a stable trait of depression was more severe in the depression group vs. healthy control.
** Childhood trauma &amp; OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model.
** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity.
** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation.
** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) &amp; WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation.
* Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy &amp; MeST, both geared towards recalling memories in detail.
* '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors.
* '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior.

=== [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] ===
Crane et al. (2016) – Community adolescent cohort

* Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16)
* Tested whether OGM predicts depression/suicidality and moderates life stress
* '''Findings:'''
** OGM did not predict depression, suicidal ideation, or self-harm
** OGM did not moderate the effect of life events
** Life events strongly predicted all outcomes
* '''Interpretation:'''
** OGM may not function as a general vulnerability factor in community samples
** Likely only relevant in high-risk or depressed populations

=== Sumner et al. (2010) – Meta-analysis (OGM &amp; depression) ===

* Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression
* '''Findings:'''
** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up
** Effect remains '''even after controlling for baseline depression'''
** Overall effect size is '''small but significant (~1–2% variance explained)'''
* '''Moderators:'''
** Stronger effect in '''clinically depressed samples vs nonclinical'''
** Stronger with '''shorter follow-up periods'''

* '''Interpretation:'''
** OGM is a '''predictor of depression maintenance''', not just a correlate
** Likely acts as a '''vulnerability factor''', especially in high-risk groups
{{Notice|OGM works under certain conditions (clinical / high-risk), not universally}}

=== Moore &amp; Zoellner (2007) – Evaluative Review (OGM &amp; trauma) ===

* Evaluative review of '''24 studies''' examining trauma exposure and OGM
* '''Core question:''' Does trauma cause overgeneral memory?
'''Findings'''
* ❌ '''No consistent link''' between trauma exposure and OGM
* ✅ OGM is more consistently linked to:
** '''Depression'''
** '''PTSD symptoms (intrusions, avoidance)'''
* Trauma exposure alone is '''not sufficient''' to produce OGM
* OGM appears more tied to '''psychopathology''', not the event itself
'''Key nuance'''
* Post-trauma '''symptoms''' (not the trauma itself) are what matter
* Evidence across studies is '''mixed and methodologically inconsistent'''
'''Interpretation'''
* Challenges the classic “trauma → OGM” theory
* Supports alternative view:&lt;blockquote&gt;OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)&lt;/blockquote&gt;
*Trauma alone isn’t enough — psychological response matters

=== Crane &amp; Duggan (2009) – CSA onset &amp; OGM in suicidal patients ===

* Clinical sample of '''49 patients with recurrent suicidal behavior'''
* Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM

'''Findings'''

* Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories)
* Effect remained after controlling for:
** depression
** verbal fluency
* Presence vs absence of CSA '''did NOT differ in OGM'''
** (because sample already highly clinical)

'''Interpretation'''

* Not trauma itself → but '''timing of trauma (early development)''' matters
* Supports idea that OGM develops when:
** memory systems are still forming
** avoidance becomes a learned coping style 

'''Insights'''

# Analysis of OGM in a suicidal population.
# Moore &amp; Zoellner had trauma alone ≠ OGM, while Crane &amp; Duggan added some trauma characteristics from early developmental trauma. 
# ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' 

=== Champagne et al. (2016) – OGM in adolescent MDD ===

* Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed)
* Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)'''

'''Findings:'''

* Adolescents with '''current AND remitted MDD''' showed more OGM than controls
* No significant difference between current vs remitted groups
* OGM present for both '''positive and negative cues'''

'''Interpretation:'''

* OGM is likely a '''trait-like vulnerability''', not just a temporary symptom
* May contribute to '''risk of recurrence''' in depression

'''1. Trait vs state question (very important)'''

This paper directly answers:&lt;blockquote&gt;Is OGM just a symptom or a vulnerability?&lt;/blockquote&gt;👉 Answer: '''likely vulnerability'''

That’s big for your argument.
----'''2. Adolescent + clinical sample'''

* Not just adults
* Not just community

👉 Stronger relevance than many papers
----'''3. Fits your mechanism pathway'''

Supports:&lt;blockquote&gt;OGM persists → increases long-term risk → can feed into suicidal ideation&lt;/blockquote&gt;'''Limitations'''

* Small sample (n=65 total)
* Cross-sectional → can’t prove causality

* Can’t tell if OGM came '''before''' depression or is a “scar”

=== Arie et al. (2008) – OGM, problem solving &amp; suicidal behavior ===

* Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls)
* Tested Williams’ model linking OGM, problem solving, and suicide

'''Findings:'''

* Suicidal adolescents showed:
** '''More OGM (less specific memory)'''
** '''Worse interpersonal problem-solving ability'''
** '''Higher hopelessness'''
* OGM significantly correlated with:
** poorer problem solving
** higher hopelessness
** greater suicidal behavior
* Negative childhood life events also linked to OGM and suicide

'''Interpretation:'''

* Supports pathway:&lt;blockquote&gt;OGM → poor problem solving → hopelessness → suicidal behavior&lt;/blockquote&gt;
* OGM may limit access to specific past experiences needed to solve problems effectively

'''Insights'''

* Study on population actually attempting suicide.
* Points out a clear pathway
* Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness

'''Limitations''' 

* n=75, small sample
* cross sectional, so causality can not be pinpointed
* inpatient sample, so cannot be generalizable

=== Kaviani et al. (2011) – OGM, problem solving &amp; suicidal ideation ===

* Clinical sample: depressed patients '''with vs without suicidal ideation'''
* Tested links between OGM, problem solving, depression, and hopelessness

'''Findings:'''

* Suicidal ideation group:
** '''Less specific autobiographical memories (more OGM)'''
** '''More hopelessness'''
** '''Less effective problem-solving strategies'''
* OGM associated with:
** poorer problem solving
** greater hopelessness
* Evidence for a “cycle”:&lt;blockquote&gt;OGM → poor problem solving → hopelessness → suicidal ideation&lt;/blockquote&gt;

'''Interpretation:'''

* OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving
* Supports a '''cognitive pathway to suicide risk'''

'''Insights'''

* Studies suicidal ideation directly.
* Mechanism displayed: OGM --&gt; problem solving --&gt; hopelessness --&gt; suicidal ideation
* Within-depression comparison

'''Limitations'''

* n=40
* Cross-sectional, lacks causality
* Specific country (Iran)

=== Zhu et al. (2025) – OGM &amp; suicidal ideation (ML study) ===

* Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels)
* Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation

'''Findings:'''

* Patients with suicidal ideation:
** '''More OGM (fewer specific memories)'''
** Especially impaired recall of '''positive memories'''
* OGM strongly associated with '''severity of suicidal ideation'''
* Crucially:
** '''OGM predicts suicidal ideation independently of depression severity'''
* ML model showed:
** OGM features were key for identifying '''presence of suicidal ideation'''
** Depression severity features ≠ sufficient to predict SI

'''Interpretation:'''

* OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression
* Supports:&lt;blockquote&gt;OGM contributes uniquely to suicidal ideation beyond general depression&lt;/blockquote&gt;'''Insights'''

* OGM is unique in contribution/specifically tied to SI.
* Separated depression anxiety and suicidal ideation.

OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation.

== Narratives? ==

== See also ==

* [[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]]

== Notes ==
{{Reflist}}</text>
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    </revision>
  </page>
  <page>
    <title>User:ThinkingScience/ND Inspired Idea Notebook</title>
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      <timestamp>2026-04-22T07:29:26Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* April 21, 2026 */ diary for today, a few notes</comment>
      <origin>2805914</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1995" sha1="i2xf7l3mnbnu7wxe4v5d3lolpwh2o2z" xml:space="preserve">Template Links:
* {{tl|Draft}}
* {{tl|underconstruction}}

'''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].'''

== &quot;Diary&quot; ==

== April 18, 2026 ==

A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a &quot;diary&quot; that will be only regarding the project.

Turned &quot;me language&quot; into expressing that everyone is welcome, that I don't &quot;own&quot; [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says &quot;I did this&quot; &quot;I did that&quot; should be gone. I think this was an improvement of some sort.

Plan for next edits on the draft page: Add an &lt;nowiki&gt;&quot; == Old Methodology needing updating == &quot;&lt;/nowiki&gt; where I will add old methodologies where I had not planned ahead too good and the &quot;Do no harm&quot; I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago.

Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future?

== April 20, 2026 ==

Why does it seem like I'm the only one using the word &quot;methodology&quot;? Did the [[Wikiversity:Research ethics]] mention it?
* I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now.

== April 21, 2026 ==

I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the &quot;Do no harm&quot;. Considering I have almost not developed anything but I still gotta work on this...to do...

== April 22, 2026 ==

I am yearning / looking forward to working on video notes in a &quot;Do no harm&quot; way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different.</text>
      <sha1>i2xf7l3mnbnu7wxe4v5d3lolpwh2o2z</sha1>
    </revision>
  </page>
  <page>
    <title>Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions</title>
    <ns>118</ns>
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    <revision>
      <id>2805891</id>
      <parentid>2805602</parentid>
      <timestamp>2026-04-22T06:41:48Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;AI Mode&quot; by Google inputs/outputs */ System of making this page &quot;reference compatible&quot; in regards to any pages which link to &quot;AI generated decision making&quot; to make work easier</comment>
      <origin>2805891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4240" sha1="8wzm09tx8udi5zbupz6xn4fujlv0ucx" xml:space="preserve">{{AI-generated}}

For practical purposes we can assume that everything on this page is &quot;AI-generated&quot;. Where it is not, it will be stated.

How questions were generated appear on this page for transparency reasons. The input and output questions are documented.

This is a page where one can get a question and then work further to refine the question. This is some disorganized inputs and outputs. Inputs are sent to an AI/LLM and outputs is about what the AI/LLM gives the user.

When it says &quot;'''human output'''&quot; that means that a user thought of a question based on an AI's/LLM's question. Simply a follow-up question to an AI's/LLM's question.

== &quot;AI Mode&quot; by Google inputs/outputs ==

Please feel free to add subsections here. 

* human input April 18, 2026: I have an idea and I'm developing the idea here on English Wikiversity: Draft:The Neurodiversity-inspired Idea. What kind of questions might I need to move forward. I'm looking for academic feedback. The problem is &quot;people don't understand themselves&quot; and have to go through &quot;dragged-out diagnosis processes&quot; by psychiatrists. The problem to be solved: how the brain works and presenting a system how it can be mapped.
** output Purpose: &quot;What specific problem does this idea solve for the reader?&quot;
** output Reach: &quot;Who is the primary 'student' for this draft?&quot;
*** human output: Who is the target audience for this draft?
** output Validation: &quot;What existing research or lived experience supports this claim?&quot;
** output Interactivity: &quot;What can a reader do with this information right now?&quot;

* human input: Is a &quot;neurotype&quot; up for debate? Does every researcher have their own definition of what a &quot;neurotype&quot; is? Based on this input can you suggest better questhe tions?

=== Methodology questions ===

* human input April 20, 2026: On Wikiversity they say &quot;ask the right questions with your idea&quot;. I got an idea, this is my idea: &quot;The Neurodiversity-inspired idea&quot;. My methodology is about watching videos of famous people but I want to do the public notes in a way that doesn't &quot;offend the famous people&quot; when I compare them with other famous people.
** output: &quot;How does this person's way of processing information differ from or similar to the other person in this comparison?&quot;, &quot;What strengths or 'superpowers' are evident in how they navigate this specific situation?&quot;. Strategy for Non-Offensive Public Notes. Focus on Behaviours, Not Diagnoses: Only comment on observed actions or statements. Use phrases like, &quot;Their speaking style is consistent with...&quot; rather than &quot;They are...&quot;.

'''Methodology questions'''

When watching a famous person in a video interview in the current &quot;methodology&quot;:
* How does this person's way of processing information differ to another person in this comparison? (modified by a human)
* What strengths or 'superpowers' are evident in how they navigate this specific situation?

==== Failures in output we can learn from in Methodology questions to AIs/LLMs ====

It's not always easy to find the right questions. When the question has some problems that causes an output that doesn't make sense they can be put in this section.

* human input April 19, 2026: I'm looking for an English-Wikiversity-compliant methodology to study famous people in video interviews while &quot;being allowed&quot; to group them together without it being &quot;heresy&quot; in scientific communities. Can you formulate that as a question?
** output &quot;didn't make sense&quot; / &quot;was irrelevant&quot; (conclusion by human contributor)

== &quot;Reference Coding&quot; ==

Here happens a &quot;reference coding&quot; so it's easy to find an input when it is being referenced from another page on Wikiversity. ie. [1] should be the same as [1] on another page which is referencing the [1] here.

These inputs/outputs were done with &quot;AI Mode&quot; by Google.
* human input April 22[1]: I need somebody who is a famous person and that we can tie to the word &quot;draft&quot; even indirectly and that is a modern person and that appears in many different video interviews.
** output: A compelling modern person for your &quot;Method development&quot; section is Taylor Swift. She is a modern icon who appears in countless video interviews and can be tied directly to the word &quot;draft&quot; through her creative process and public statements.</text>
      <sha1>8wzm09tx8udi5zbupz6xn4fujlv0ucx</sha1>
    </revision>
    <revision>
      <id>2805892</id>
      <parentid>2805891</parentid>
      <timestamp>2026-04-22T06:42:48Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;Reference Coding&quot; */ I am not sure how to do this but this is the best I can do now so I can continue my work</comment>
      <origin>2805892</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4254" sha1="48a7zcjhzyuor7ndcwpzpcod5jo1qhi" xml:space="preserve">{{AI-generated}}

For practical purposes we can assume that everything on this page is &quot;AI-generated&quot;. Where it is not, it will be stated.

How questions were generated appear on this page for transparency reasons. The input and output questions are documented.

This is a page where one can get a question and then work further to refine the question. This is some disorganized inputs and outputs. Inputs are sent to an AI/LLM and outputs is about what the AI/LLM gives the user.

When it says &quot;'''human output'''&quot; that means that a user thought of a question based on an AI's/LLM's question. Simply a follow-up question to an AI's/LLM's question.

== &quot;AI Mode&quot; by Google inputs/outputs ==

Please feel free to add subsections here. 

* human input April 18, 2026: I have an idea and I'm developing the idea here on English Wikiversity: Draft:The Neurodiversity-inspired Idea. What kind of questions might I need to move forward. I'm looking for academic feedback. The problem is &quot;people don't understand themselves&quot; and have to go through &quot;dragged-out diagnosis processes&quot; by psychiatrists. The problem to be solved: how the brain works and presenting a system how it can be mapped.
** output Purpose: &quot;What specific problem does this idea solve for the reader?&quot;
** output Reach: &quot;Who is the primary 'student' for this draft?&quot;
*** human output: Who is the target audience for this draft?
** output Validation: &quot;What existing research or lived experience supports this claim?&quot;
** output Interactivity: &quot;What can a reader do with this information right now?&quot;

* human input: Is a &quot;neurotype&quot; up for debate? Does every researcher have their own definition of what a &quot;neurotype&quot; is? Based on this input can you suggest better questhe tions?

=== Methodology questions ===

* human input April 20, 2026: On Wikiversity they say &quot;ask the right questions with your idea&quot;. I got an idea, this is my idea: &quot;The Neurodiversity-inspired idea&quot;. My methodology is about watching videos of famous people but I want to do the public notes in a way that doesn't &quot;offend the famous people&quot; when I compare them with other famous people.
** output: &quot;How does this person's way of processing information differ from or similar to the other person in this comparison?&quot;, &quot;What strengths or 'superpowers' are evident in how they navigate this specific situation?&quot;. Strategy for Non-Offensive Public Notes. Focus on Behaviours, Not Diagnoses: Only comment on observed actions or statements. Use phrases like, &quot;Their speaking style is consistent with...&quot; rather than &quot;They are...&quot;.

'''Methodology questions'''

When watching a famous person in a video interview in the current &quot;methodology&quot;:
* How does this person's way of processing information differ to another person in this comparison? (modified by a human)
* What strengths or 'superpowers' are evident in how they navigate this specific situation?

==== Failures in output we can learn from in Methodology questions to AIs/LLMs ====

It's not always easy to find the right questions. When the question has some problems that causes an output that doesn't make sense they can be put in this section.

* human input April 19, 2026: I'm looking for an English-Wikiversity-compliant methodology to study famous people in video interviews while &quot;being allowed&quot; to group them together without it being &quot;heresy&quot; in scientific communities. Can you formulate that as a question?
** output &quot;didn't make sense&quot; / &quot;was irrelevant&quot; (conclusion by human contributor)

== &quot;Reference Coding&quot; ==

Here happens a &quot;reference coding&quot; so it's easy to find an input when it is being referenced from another page on Wikiversity. ie. [1] should be the same as [1] on another page which is referencing the [1] here.

These inputs/outputs were done with &quot;AI Mode&quot; by Google.

* Reference [1]
* human input April 22: I need somebody who is a famous person and that we can tie to the word &quot;draft&quot; even indirectly and that is a modern person and that appears in many different video interviews.
** output: A compelling modern person for your &quot;Method development&quot; section is Taylor Swift. She is a modern icon who appears in countless video interviews and can be tied directly to the word &quot;draft&quot; through her creative process and public statements.</text>
      <sha1>48a7zcjhzyuor7ndcwpzpcod5jo1qhi</sha1>
    </revision>
    <revision>
      <id>2805898</id>
      <parentid>2805892</parentid>
      <timestamp>2026-04-22T06:53:45Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;Reference Coding&quot; */ Making a warning. Links may break if the section's name is changed.</comment>
      <origin>2805898</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4466" sha1="84832duwnllnp9sd1aai2k34iw1cffv" xml:space="preserve">{{AI-generated}}

For practical purposes we can assume that everything on this page is &quot;AI-generated&quot;. Where it is not, it will be stated.

How questions were generated appear on this page for transparency reasons. The input and output questions are documented.

This is a page where one can get a question and then work further to refine the question. This is some disorganized inputs and outputs. Inputs are sent to an AI/LLM and outputs is about what the AI/LLM gives the user.

When it says &quot;'''human output'''&quot; that means that a user thought of a question based on an AI's/LLM's question. Simply a follow-up question to an AI's/LLM's question.

== &quot;AI Mode&quot; by Google inputs/outputs ==

Please feel free to add subsections here. 

* human input April 18, 2026: I have an idea and I'm developing the idea here on English Wikiversity: Draft:The Neurodiversity-inspired Idea. What kind of questions might I need to move forward. I'm looking for academic feedback. The problem is &quot;people don't understand themselves&quot; and have to go through &quot;dragged-out diagnosis processes&quot; by psychiatrists. The problem to be solved: how the brain works and presenting a system how it can be mapped.
** output Purpose: &quot;What specific problem does this idea solve for the reader?&quot;
** output Reach: &quot;Who is the primary 'student' for this draft?&quot;
*** human output: Who is the target audience for this draft?
** output Validation: &quot;What existing research or lived experience supports this claim?&quot;
** output Interactivity: &quot;What can a reader do with this information right now?&quot;

* human input: Is a &quot;neurotype&quot; up for debate? Does every researcher have their own definition of what a &quot;neurotype&quot; is? Based on this input can you suggest better questhe tions?

=== Methodology questions ===

* human input April 20, 2026: On Wikiversity they say &quot;ask the right questions with your idea&quot;. I got an idea, this is my idea: &quot;The Neurodiversity-inspired idea&quot;. My methodology is about watching videos of famous people but I want to do the public notes in a way that doesn't &quot;offend the famous people&quot; when I compare them with other famous people.
** output: &quot;How does this person's way of processing information differ from or similar to the other person in this comparison?&quot;, &quot;What strengths or 'superpowers' are evident in how they navigate this specific situation?&quot;. Strategy for Non-Offensive Public Notes. Focus on Behaviours, Not Diagnoses: Only comment on observed actions or statements. Use phrases like, &quot;Their speaking style is consistent with...&quot; rather than &quot;They are...&quot;.

'''Methodology questions'''

When watching a famous person in a video interview in the current &quot;methodology&quot;:
* How does this person's way of processing information differ to another person in this comparison? (modified by a human)
* What strengths or 'superpowers' are evident in how they navigate this specific situation?

==== Failures in output we can learn from in Methodology questions to AIs/LLMs ====

It's not always easy to find the right questions. When the question has some problems that causes an output that doesn't make sense they can be put in this section.

* human input April 19, 2026: I'm looking for an English-Wikiversity-compliant methodology to study famous people in video interviews while &quot;being allowed&quot; to group them together without it being &quot;heresy&quot; in scientific communities. Can you formulate that as a question?
** output &quot;didn't make sense&quot; / &quot;was irrelevant&quot; (conclusion by human contributor)

== &quot;Reference Coding&quot; ==

'''⚠️⚠️⚠️ WARNING ⚠️⚠️⚠️: Very important. This section's name should not be changed because other pages already link here and if the section's name is changed those links will break!
'''

Here happens a &quot;reference coding&quot; so it's easy to find an input when it is being referenced from another page on Wikiversity. ie. [1] should be the same as [1] on another page which is referencing the [1] here.

These inputs/outputs were done with &quot;AI Mode&quot; by Google.

* Reference [1]
* human input April 22: I need somebody who is a famous person and that we can tie to the word &quot;draft&quot; even indirectly and that is a modern person and that appears in many different video interviews.
** output: A compelling modern person for your &quot;Method development&quot; section is Taylor Swift. She is a modern icon who appears in countless video interviews and can be tied directly to the word &quot;draft&quot; through her creative process and public statements.</text>
      <sha1>84832duwnllnp9sd1aai2k34iw1cffv</sha1>
    </revision>
    <revision>
      <id>2805899</id>
      <parentid>2805898</parentid>
      <timestamp>2026-04-22T06:54:47Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* &quot;Reference Coding&quot; */ The warning may be deterring contributors so if there are any questions they are now advised to use the talk page for very specific questions</comment>
      <origin>2805899</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4510" sha1="o9d790xmxx0w9bixzxjfso5y7s7dxl8" xml:space="preserve">{{AI-generated}}

For practical purposes we can assume that everything on this page is &quot;AI-generated&quot;. Where it is not, it will be stated.

How questions were generated appear on this page for transparency reasons. The input and output questions are documented.

This is a page where one can get a question and then work further to refine the question. This is some disorganized inputs and outputs. Inputs are sent to an AI/LLM and outputs is about what the AI/LLM gives the user.

When it says &quot;'''human output'''&quot; that means that a user thought of a question based on an AI's/LLM's question. Simply a follow-up question to an AI's/LLM's question.

== &quot;AI Mode&quot; by Google inputs/outputs ==

Please feel free to add subsections here. 

* human input April 18, 2026: I have an idea and I'm developing the idea here on English Wikiversity: Draft:The Neurodiversity-inspired Idea. What kind of questions might I need to move forward. I'm looking for academic feedback. The problem is &quot;people don't understand themselves&quot; and have to go through &quot;dragged-out diagnosis processes&quot; by psychiatrists. The problem to be solved: how the brain works and presenting a system how it can be mapped.
** output Purpose: &quot;What specific problem does this idea solve for the reader?&quot;
** output Reach: &quot;Who is the primary 'student' for this draft?&quot;
*** human output: Who is the target audience for this draft?
** output Validation: &quot;What existing research or lived experience supports this claim?&quot;
** output Interactivity: &quot;What can a reader do with this information right now?&quot;

* human input: Is a &quot;neurotype&quot; up for debate? Does every researcher have their own definition of what a &quot;neurotype&quot; is? Based on this input can you suggest better questhe tions?

=== Methodology questions ===

* human input April 20, 2026: On Wikiversity they say &quot;ask the right questions with your idea&quot;. I got an idea, this is my idea: &quot;The Neurodiversity-inspired idea&quot;. My methodology is about watching videos of famous people but I want to do the public notes in a way that doesn't &quot;offend the famous people&quot; when I compare them with other famous people.
** output: &quot;How does this person's way of processing information differ from or similar to the other person in this comparison?&quot;, &quot;What strengths or 'superpowers' are evident in how they navigate this specific situation?&quot;. Strategy for Non-Offensive Public Notes. Focus on Behaviours, Not Diagnoses: Only comment on observed actions or statements. Use phrases like, &quot;Their speaking style is consistent with...&quot; rather than &quot;They are...&quot;.

'''Methodology questions'''

When watching a famous person in a video interview in the current &quot;methodology&quot;:
* How does this person's way of processing information differ to another person in this comparison? (modified by a human)
* What strengths or 'superpowers' are evident in how they navigate this specific situation?

==== Failures in output we can learn from in Methodology questions to AIs/LLMs ====

It's not always easy to find the right questions. When the question has some problems that causes an output that doesn't make sense they can be put in this section.

* human input April 19, 2026: I'm looking for an English-Wikiversity-compliant methodology to study famous people in video interviews while &quot;being allowed&quot; to group them together without it being &quot;heresy&quot; in scientific communities. Can you formulate that as a question?
** output &quot;didn't make sense&quot; / &quot;was irrelevant&quot; (conclusion by human contributor)

== &quot;Reference Coding&quot; ==

'''⚠️⚠️⚠️ WARNING ⚠️⚠️⚠️: Very important. This section's name should not be changed because other pages already link here and if the section's name is changed those links will break! For any questions please use the talk page.
'''

Here happens a &quot;reference coding&quot; so it's easy to find an input when it is being referenced from another page on Wikiversity. ie. [1] should be the same as [1] on another page which is referencing the [1] here.

These inputs/outputs were done with &quot;AI Mode&quot; by Google.

* Reference [1]
* human input April 22: I need somebody who is a famous person and that we can tie to the word &quot;draft&quot; even indirectly and that is a modern person and that appears in many different video interviews.
** output: A compelling modern person for your &quot;Method development&quot; section is Taylor Swift. She is a modern icon who appears in countless video interviews and can be tied directly to the word &quot;draft&quot; through her creative process and public statements.</text>
      <sha1>o9d790xmxx0w9bixzxjfso5y7s7dxl8</sha1>
    </revision>
  </page>
  <page>
    <title>Operating Systems (Hands-On)/Implementing a File System/Review Quiz</title>
    <ns>0</ns>
    <id>329206</id>
    <revision>
      <id>2805748</id>
      <parentid>2805732</parentid>
      <timestamp>2026-04-21T12:40:35Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>added [[Category:Quiz]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2805748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2676" sha1="5bx0vvvjlqrwtnzlytbv31nsjh8j7gk" xml:space="preserve">== Ext3 Review Quiz ==
&lt;quiz display=simple shuffleanswers=true&gt;

{ Which field is conspicuously absent from an ext3 inode?
| type=&quot;()&quot;}
+ Filename
- File size
- Owner user ID
- Modification timestamp

{ In ext3, what does a '0' bit in the block bitmap indicate?
| type=&quot;()&quot;}
+ The block is free
- The block is allocated
- The block has been corrupted
- The block belongs to the inode table

{ With 4 KB blocks, how much data can a single ext3 inode address using only its 12 direct block pointers?
| type=&quot;()&quot;}
+ 48 KB
- 4 MB
- 4 GB
- 12 KB, one block per pointer, obviously

{ By convention, which inode number always holds the root directory in ext3?
| type=&quot;()&quot;}
+ Inode 2
- Inode 0
- Inode 1
- Whichever inode the superblock points to that day

{ What problem did journaling solve when ext3 succeeded ext2?
| type=&quot;()&quot;}
+ The need to run fsck after every unclean shutdown
- Fragmentation of large files across block groups
- The 2 GB file size limit
- The lack of support for symbolic links

{ A filesystem formatted for large files may run out of inodes before it runs out of disk space. What does this tell you about inode-based systems?
| type=&quot;()&quot;}
+ The maximum number of files is fixed at format time
- Inodes grow dynamically as files are added
- Large files consume more inodes than small files
- The superblock is responsible for allocating new inodes on demand

{ Why is FAT32 still commonly used on removable media like camera memory cards, despite its known weaknesses?
| type=&quot;()&quot;}
+ Its file allocation table grows dynamically, making it well-suited to highly variable file counts
- It has better crash recovery than ext3
- It supports longer filenames than ext3
- Camera manufacturers find the linked-list structure more aesthetically pleasing

{ In ext3, a single indirect block pointer with 4 KB blocks can address approximately how much data?
| type=&quot;()&quot;}
+ 4 MB
- 48 KB
- 4 GB
- 4 TB

{ What is the key advantage of ext3's block group structure over treating the entire disk as one flat space?
| type=&quot;()&quot;}
+ It keeps a file's inodes and data blocks physically close together, improving locality
- It allows the filesystem to support more than 4 GB of total storage
- It eliminates the need for a superblock backup
- It makes the block bitmap unnecessary

{ In an ext3 inode system, what is a hard link?
| type=&quot;()&quot;}
+ Two directory entries with different names pointing to the same inode number
- A special inode type that stores a path string to another file
- A copy of a file stored in a separate block group for redundancy
- An entry in the journal that records a previous write operation

&lt;/quiz&gt;

[[Category:Operating systems]]
[[Category:Quiz]]</text>
      <sha1>5bx0vvvjlqrwtnzlytbv31nsjh8j7gk</sha1>
    </revision>
  </page>
  <page>
    <title>File:VLSI.Arith.2A.CLA.20260421.pdf</title>
    <ns>6</ns>
    <id>329208</id>
    <revision>
      <id>2805762</id>
      <timestamp>2026-04-21T13:35:22Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Carry Lookahead Adders 2A traditional (20260421 - 20260420)
|Source={{own|Young1lim}}
|Date=2026-04-21
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2805762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="282" sha1="g6n0oh3cpyipu1jykqotzr618wiapng" xml:space="preserve">== Summary ==
{{Information
|Description=Carry Lookahead Adders 2A traditional (20260421 - 20260420)
|Source={{own|Young1lim}}
|Date=2026-04-21
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>g6n0oh3cpyipu1jykqotzr618wiapng</sha1>
    </revision>
  </page>
  <page>
    <title>File:C04.SA0.PtrOperator.1A.20260421.pdf</title>
    <ns>6</ns>
    <id>329209</id>
    <revision>
      <id>2805772</id>
      <timestamp>2026-04-21T13:50:41Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=C04.SA0: Address and Dereference Operators (20260421 - 20260420)
|Source={{own|Young1lim}}
|Date=2026-04-21
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2805772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="286" sha1="7r5h3lpfh99utmtg5w6pebn7tzwl6a9" xml:space="preserve">== Summary ==
{{Information
|Description=C04.SA0: Address and Dereference Operators (20260421 - 20260420)
|Source={{own|Young1lim}}
|Date=2026-04-21
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>7r5h3lpfh99utmtg5w6pebn7tzwl6a9</sha1>
    </revision>
  </page>
  <page>
    <title>Draft:The Neurodiversity-inspired Idea/Method development through video notes</title>
    <ns>118</ns>
    <id>329210</id>
    <revision>
      <id>2805774</id>
      <timestamp>2026-04-21T13:57:29Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>first draft</comment>
      <origin>2805774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2380" sha1="hi8aei7rrbv2bchtab2ns1txadzfdci" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people like human beings. How one expresses oneself is important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words &quot;draft, neurodiversity, inspiration or idea&quot; which is how this start is made. This idea may be lacking but one starts, makes mistakes, then later we learn how to do it better and then we fix those mistakes.</text>
      <sha1>hi8aei7rrbv2bchtab2ns1txadzfdci</sha1>
    </revision>
    <revision>
      <id>2805873</id>
      <parentid>2805774</parentid>
      <timestamp>2026-04-22T06:03:20Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Questions */ clarification</comment>
      <origin>2805873</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2400" sha1="m5sqnv33syjfp3u5nuimre7gnkh2nq1" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people like human beings. How one expresses oneself is important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words &quot;draft, neurodiversity, inspiration or idea&quot; which is how this start is made. This idea may be lacking but one starts, makes mistakes, then later we learn how to do it better and then we fix those mistakes.</text>
      <sha1>m5sqnv33syjfp3u5nuimre7gnkh2nq1</sha1>
    </revision>
    <revision>
      <id>2805879</id>
      <parentid>2805873</parentid>
      <timestamp>2026-04-22T06:09:53Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; */ after analyzing the draft this seems to be the first section that focus  on &quot;the work&quot;</comment>
      <origin>2805879</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2417" sha1="ooaens8uqni6wjvrtjjvo0opyqhkk5i" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people like human beings. How one expresses oneself is important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words &quot;draft, neurodiversity, inspiration or idea&quot; which is how this start is made. This idea may be lacking but one starts, makes mistakes, then later we learn how to do it better and then we fix those mistakes.</text>
      <sha1>ooaens8uqni6wjvrtjjvo0opyqhkk5i</sha1>
    </revision>
    <revision>
      <id>2805884</id>
      <parentid>2805879</parentid>
      <timestamp>2026-04-22T06:14:51Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; */ for me a small change</comment>
      <origin>2805884</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2423" sha1="8zz2wzupjwyao046vcjzgl5y0tpubng" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words &quot;draft, neurodiversity, inspiration or idea&quot; which is how this start is made. This idea may be lacking but one starts, makes mistakes, then later we learn how to do it better and then we fix those mistakes.</text>
      <sha1>8zz2wzupjwyao046vcjzgl5y0tpubng</sha1>
    </revision>
    <revision>
      <id>2805886</id>
      <parentid>2805884</parentid>
      <timestamp>2026-04-22T06:16:47Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; */ small quick fixes and clarifications during these small edits</comment>
      <origin>2805886</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2425" sha1="o5f9z4zo1d713kik7aohxqvace6u20l" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.</text>
      <sha1>o5f9z4zo1d713kik7aohxqvace6u20l</sha1>
    </revision>
    <revision>
      <id>2805896</id>
      <parentid>2805886</parentid>
      <timestamp>2026-04-22T06:48:42Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; */ complexity may be/is increasing</comment>
      <origin>2805896</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3837" sha1="d7p8ryrn88dwkgzh92d3ys49g8tbk2s" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.

== Choosing video material and video notes can wait... ==

Why can video notes wait? The first task is choosing the video material and explaining why the specific material was chosen. It may not always be easy to take a decision and ie. ChatGPT or &quot;AI Mode&quot; can sometimes be used to make it easier to just choose something so one can focus on work and less on decisions. For some students/contributors that may be helpful.

The video notes though may be done step by step, one chooses a video material, explains the reasoning behind it and then focuses on working on the video notes in a &quot;Do no harm&quot; manner.

* How to associate the first person we focus on regarding the word draft? Using &quot;AI Mode&quot; by Google on April 22, 2026, as a way to make &quot;decisions easier&quot; this input helped this project moving forward with the first &quot;choice&quot;: Taylor Swift[1]

== How the Linking References System works ==

ie. [1], [2] etc.

This section may be superfluous once a better system has replaced this.

Purpose: teach the students/contributors how to link to a central place where human generated inputs are made and &quot;AI-generated&quot; outputs results are made.

When you see ie. a [1] that means that

=== References/in-Wikiversity-Wikilinks ===

'''Most importantly''':

All in-Wikiversity references like ie. [1] can be found at [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]</text>
      <sha1>d7p8ryrn88dwkgzh92d3ys49g8tbk2s</sha1>
    </revision>
    <revision>
      <id>2805897</id>
      <parentid>2805896</parentid>
      <timestamp>2026-04-22T06:51:36Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* How the Linking References System works */ Finishing an explanation: This is challenging for me, I may need your help. I just want to work but I also have to organize the information</comment>
      <origin>2805897</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4163" sha1="bchg63ilux62t66h96rx6ojcxlhf3ss" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.

== Choosing video material and video notes can wait... ==

Why can video notes wait? The first task is choosing the video material and explaining why the specific material was chosen. It may not always be easy to take a decision and ie. ChatGPT or &quot;AI Mode&quot; can sometimes be used to make it easier to just choose something so one can focus on work and less on decisions. For some students/contributors that may be helpful.

The video notes though may be done step by step, one chooses a video material, explains the reasoning behind it and then focuses on working on the video notes in a &quot;Do no harm&quot; manner.

* How to associate the first person we focus on regarding the word draft? Using &quot;AI Mode&quot; by Google on April 22, 2026, as a way to make &quot;decisions easier&quot; this input helped this project moving forward with the first &quot;choice&quot;: Taylor Swift[1]

== How the Linking References System works ==

ie. [1], [2] etc.

This section may be superfluous once a better system has replaced this.

Purpose: teach the students/contributors how to link to a central place where human generated inputs are made and &quot;AI-generated&quot; outputs results are made.

When you see ie. a [1] that means that you can find that [1] input made by a human and output most probably made by a an &quot;AI program&quot; or Large Language Model here if you look hard enough. Otherwise please submit feedback for how the user experience was. Here's the page: [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]

=== References/in-Wikiversity-Wikilinks ===

'''Most importantly''':

All in-Wikiversity references like ie. [1] can be found at [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]</text>
      <sha1>bchg63ilux62t66h96rx6ojcxlhf3ss</sha1>
    </revision>
    <revision>
      <id>2805900</id>
      <parentid>2805897</parentid>
      <timestamp>2026-04-22T06:56:33Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Choosing video material and video notes can wait... */ title changed to == Choosing video material == and explanations added. Was confusing even for me</comment>
      <origin>2805900</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4204" sha1="9q5pmkogzsxb3siyoxc73yctsxttq9b" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.

== Choosing video material ==

Video notes can wait. Let's focus first on choosing video material.

Why can video notes wait? The first task is choosing the video material and explaining why the specific material was chosen. It may not always be easy to take a decision and ie. ChatGPT or &quot;AI Mode&quot; can sometimes be used to make it easier to just choose something so one can focus on work and less on decisions. For some students/contributors that may be helpful.

The video notes though may be done step by step, one chooses a video material, explains the reasoning behind it and then focuses on working on the video notes in a &quot;Do no harm&quot; manner.

* How to associate the first person we focus on regarding the word draft? Using &quot;AI Mode&quot; by Google on April 22, 2026, as a way to make &quot;decisions easier&quot; this input helped this project moving forward with the first &quot;choice&quot;: Taylor Swift[1]

== How the Linking References System works ==

ie. [1], [2] etc.

This section may be superfluous once a better system has replaced this.

Purpose: teach the students/contributors how to link to a central place where human generated inputs are made and &quot;AI-generated&quot; outputs results are made.

When you see ie. a [1] that means that you can find that [1] input made by a human and output most probably made by a an &quot;AI program&quot; or Large Language Model here if you look hard enough. Otherwise please submit feedback for how the user experience was. Here's the page: [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]

=== References/in-Wikiversity-Wikilinks ===

'''Most importantly''':

All in-Wikiversity references like ie. [1] can be found at [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]</text>
      <sha1>9q5pmkogzsxb3siyoxc73yctsxttq9b</sha1>
    </revision>
    <revision>
      <id>2805921</id>
      <parentid>2805900</parentid>
      <timestamp>2026-04-22T07:57:58Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Choosing video material */ I take responsibility and I encourage other contributors to do that if they want/choose to</comment>
      <origin>2805921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5411" sha1="34t5p0gv83x33k103ihquscwy2evoey" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.

== Choosing video material ==

Video notes can wait. Let's focus first on choosing video material.

Why can video notes wait? The first task is choosing the video material and explaining why the specific material was chosen. It may not always be easy to take a decision and ie. ChatGPT or &quot;AI Mode&quot; can sometimes be used to make it easier to just choose something so one can focus on work and less on decisions. For some students/contributors that may be helpful.

The video notes though may be done step by step, one chooses a video material, explains the reasoning behind it and then focuses on working on the video notes in a &quot;Do no harm&quot; manner.

* How to associate the first person we focus on regarding the word ''draft''? Using &quot;AI Mode&quot; by Google on April 22, 2026, as a way to make &quot;decisions easier&quot; this input helped this project moving forward(feedback/comments appreciated) with the first &quot;choice&quot;: 
** Taylor Swift[1]

Taylor Swift is associated with the word ''draft'' in that she has been open in videos about her music creation process.

That makes this project focus on Taylor Swift.

=== Responsibility in regards to choosing video material ===

In this section if you feel free like listing yourself feel free to. Adhering to &quot;Do no harm&quot; can be very challenging. Though we have to try our best to achieve this.

It may be good or even necessary to explain why a choice was made so that anyone joining this project can be sure that there was a 100% human judgement made when choosing a subject. As this project deals with real human beings similar to how psychiatry deals with real human beings when the Autism Spectru, the Bipolar Spectrum and similar we chosen and worked on, it dealt with real people who have feelings and can be easily offended if the &quot;wrong things&quot; are written.

This is a section where a user who is a contributor would like to take responsibility for a choice:
* [[User:ThinkingScience]]. I hereby take responsibility for the following choices: Taylor Swift. Read on my user page more at [[User:ThinkingScience#T]] why I made this choice.
* Example user 2

== How the Linking References System works ==

ie. [1], [2] etc.

This section may be superfluous once a better system has replaced this.

Purpose: teach the students/contributors how to link to a central place where human generated inputs are made and &quot;AI-generated&quot; outputs results are made.

When you see ie. a [1] that means that you can find that [1] input made by a human and output most probably made by a an &quot;AI program&quot; or Large Language Model here if you look hard enough. Otherwise please submit feedback for how the user experience was. Here's the page: [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]

=== References/in-Wikiversity-Wikilinks ===

'''Most importantly''':

All in-Wikiversity references like ie. [1] can be found at [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]</text>
      <sha1>34t5p0gv83x33k103ihquscwy2evoey</sha1>
    </revision>
    <revision>
      <id>2805922</id>
      <parentid>2805921</parentid>
      <timestamp>2026-04-22T08:03:57Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Responsibility in regards to choosing video material */ Clarification. Thinking...minimum age for famous people this entire &quot;idea&quot; examines?</comment>
      <origin>2805922</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5623" sha1="05umt2z00zvm9nurqtvwsrd91asqbjj" xml:space="preserve">{{AI-generated}}

== What {{tl|AI-generated}} means on this page ==

It means that the content on this page may be generated in the future. This is merely to &quot;future-proof&quot; this page as it may be assumed it is inevitable that one day even the content might be auto-generated and this notice existing might help students/contributors to let them know that there is a current &quot;influence&quot; by &quot;AI programs&quot; on this page, if not by content at least by influencing decisions.

If any content is generated contributors are encouraged to edit this section and make the appropriate changes to let students know that there has been a change.

What it means for now is that the contributors to this page are encouraged to interact with an &quot;AI&quot;/LLM in regards to 
* decision-making on what to focus on next that this page may need
* brainstorming with the &quot;AI program&quot; of their choice regarding what to contribute next
* ask &quot;AI program&quot; for advice on how to contribute video notes in a [[Wikiversity:Research ethics]]-way and a &quot;Do no harm&quot;/&quot;Primum non nocere&quot; way.

== Original Intent ==

The intent for this page is for it to be used to develop a 'method' for working with &quot;materials&quot; like video interviews of famous people or video interviews/content of volunteers who sign up for [[Draft:The Neurodiversity-inspired Idea]].

== Contributor/Student Questions ==

If a contributor doesn't feel to edit this page directly but the contributor/student wants to ask a question to encourage development or they have any other question, this section is for that. '''Remember: No question is stupid!'''

== Beginning work - Video notes adhering to &quot;Do no harm&quot;/&quot;Primum non nocere&quot; ==

This is the section for main content for this page which is video notes. The way one makes video notes on this page may set the example for the method that will be developed in the future.

Goal is to treat famous people like any other people: like human beings. How one expresses oneself is very important and the original creator of this page may not be the best in that area, thus '''feel free to be bold and edit this page'''.

The start is usually the hardest and based on the idea name one can focus on any famous person related to the words 
* draft
* neurodiversity
* inspiration
* idea

This is how this start is made. This idea may be lacking but one starts making mistakes, then later we learn how to do it better and then we fix those mistakes.

== Choosing video material ==

Video notes can wait. Let's focus first on choosing video material.

Why can video notes wait? The first task is choosing the video material and explaining why the specific material was chosen. It may not always be easy to take a decision and ie. ChatGPT or &quot;AI Mode&quot; can sometimes be used to make it easier to just choose something so one can focus on work and less on decisions. For some students/contributors that may be helpful.

The video notes though may be done step by step, one chooses a video material, explains the reasoning behind it and then focuses on working on the video notes in a &quot;Do no harm&quot; manner.

* How to associate the first person we focus on regarding the word ''draft''? Using &quot;AI Mode&quot; by Google on April 22, 2026, as a way to make &quot;decisions easier&quot; this input helped this project moving forward(feedback/comments appreciated) with the first &quot;choice&quot;: 
** Taylor Swift[1]

Taylor Swift is associated with the word ''draft'' in that she has been open in videos about her music creation process.

That makes this project focus on Taylor Swift.

=== Responsibility in regards to choosing video material ===

In this section if you feel free like listing yourself feel free to. Adhering to &quot;Do no harm&quot; can be very challenging. Though we have to try our best to achieve this.

It may be good or even necessary to explain why a choice was made so that anyone joining this project can be sure that there was a 100% human judgement made when choosing a subject. 

As this project deals with real human beings similar to how psychiatry deals with real human beings with ie. the Autism Spectrum, the Bipolar Spectrum and similar were chosen and worked on, it dealt with real people. Psychiatry though dealt/deal mainly with children and this project deals mainly with adults. Preferably adults whose brains have fully matured and who have lived a few years with a fully matured brain which may mean 29 years of age which may be a good minimum age for this project.

This is a section where a user who is a contributor would like to take responsibility for a choice:
* [[User:ThinkingScience]]. I hereby take responsibility for the following choices: Taylor Swift. Read on my user page more at [[User:ThinkingScience#T]] why I made this choice.
* Example user 2

== How the Linking References System works ==

ie. [1], [2] etc.

This section may be superfluous once a better system has replaced this.

Purpose: teach the students/contributors how to link to a central place where human generated inputs are made and &quot;AI-generated&quot; outputs results are made.

When you see ie. a [1] that means that you can find that [1] input made by a human and output most probably made by a an &quot;AI program&quot; or Large Language Model here if you look hard enough. Otherwise please submit feedback for how the user experience was. Here's the page: [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]

=== References/in-Wikiversity-Wikilinks ===

'''Most importantly''':

All in-Wikiversity references like ie. [1] can be found at [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#&quot;Reference_Coding&quot;]]</text>
      <sha1>05umt2z00zvm9nurqtvwsrd91asqbjj</sha1>
    </revision>
  </page>
  <page>
    <title>File:Laurent.5.Permutation.6C.20260421.pdf</title>
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        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Laurent.5: Permutation 6C (20260421 - 20260420)
|Source={{own|Young1lim}}
|Date=2026-04-21
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
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